# Real Quadratic Equation Foundations

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261003-r04-quadratic-equation-foundations · 2026-10-03 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Validity of a Quadratic-Equation Form

Over the reals, is (-2)x²+(-12)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0001/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 2. Validity of a Quadratic-Equation Form

Over the reals, is (-1)x²+(-11)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0002/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 3. Validity of a Quadratic-Equation Form

Over the reals, is (0)x²+(-10)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationNo

The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0003/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 4. Validity of a Quadratic-Equation Form

Over the reals, is (1)x²+(-9)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0004/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 5. Validity of a Quadratic-Equation Form

Over the reals, is (2)x²+(-8)x+(1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0005/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 6. Validity of a Quadratic-Equation Form

Over the reals, is (-2)x²+(-7)x+(2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0006/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 7. Validity of a Quadratic-Equation Form

Over the reals, is (-1)x²+(-6)x+(3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0007/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 8. Validity of a Quadratic-Equation Form

Over the reals, is (0)x²+(-5)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationNo

The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0008/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 9. Validity of a Quadratic-Equation Form

Over the reals, is (1)x²+(-4)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0009/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 10. Validity of a Quadratic-Equation Form

Over the reals, is (2)x²+(-3)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0010/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 11. Validity of a Quadratic-Equation Form

Over the reals, is (-2)x²+(-2)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0011/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 12. Validity of a Quadratic-Equation Form

Over the reals, is (-1)x²+(-1)x+(1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0012/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 13. Validity of a Quadratic-Equation Form

Over the reals, is (0)x²+(0)x+(2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationNo

The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0013/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 14. Validity of a Quadratic-Equation Form

Over the reals, is (1)x²+(1)x+(3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0014/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 15. Validity of a Quadratic-Equation Form

Over the reals, is (2)x²+(2)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0015/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 16. Validity of a Quadratic-Equation Form

Over the reals, is (-2)x²+(3)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0016/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 17. Validity of a Quadratic-Equation Form

Over the reals, is (-1)x²+(4)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0017/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 18. Validity of a Quadratic-Equation Form

Over the reals, is (0)x²+(5)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationNo

The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0018/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 19. Validity of a Quadratic-Equation Form

Over the reals, is (1)x²+(6)x+(1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0019/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 20. Validity of a Quadratic-Equation Form

Over the reals, is (2)x²+(7)x+(2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0020/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 21. Validity of a Quadratic-Equation Form

Over the reals, is (-2)x²+(8)x+(3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0021/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 22. Validity of a Quadratic-Equation Form

Over the reals, is (-1)x²+(9)x+(-3)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=-1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0022/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 23. Validity of a Quadratic-Equation Form

Over the reals, is (0)x²+(10)x+(-2)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationNo

The quadratic coefficient is a=0, so the answer is No. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0023/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 24. Validity of a Quadratic-Equation Form

Over the reals, is (1)x²+(11)x+(-1)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=1, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0024/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 25. Validity of a Quadratic-Equation Form

Over the reals, is (2)x²+(12)x+(0)=0 a quadratic equation in one variable? Answer Yes or No.

View answer and explanationYes

The quadratic coefficient is a=2, so the answer is Yes. A quadratic equation requires a≠0, even if a zero-coefficient x² term is written.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0025/)

Quadratic equations over the reals · qf-quadratic-form-validity · candidate





## 26. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(0)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0026/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 27. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(0)x+(0)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0027/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 28. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(0)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=4; distinct real roots=2

D=4; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0028/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 29. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(2)x+(2)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0029/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 30. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(2)x+(1)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0030/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 31. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(1)x+(2)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=9; distinct real roots=2

D=9; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0031/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 32. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(4)x+(5)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0032/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 33. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(4)x+(4)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0033/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 34. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(2)x+(3)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=16; distinct real roots=2

D=16; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0034/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 35. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(6)x+(10)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0035/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 36. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(6)x+(9)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0036/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 37. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(3)x+(4)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=25; distinct real roots=2

D=25; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0037/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 38. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(8)x+(17)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0038/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 39. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(8)x+(16)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0039/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 40. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(4)x+(5)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=36; distinct real roots=2

D=36; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0040/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 41. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(10)x+(26)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0041/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 42. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(10)x+(25)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0042/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 43. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(5)x+(6)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=49; distinct real roots=2

D=49; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0043/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 44. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(12)x+(37)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0044/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 45. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(12)x+(36)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0045/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 46. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(6)x+(7)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=64; distinct real roots=2

D=64; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0046/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 47. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(14)x+(50)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0047/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 48. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(14)x+(49)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=0; distinct real roots=1

D=0; distinct real roots=1. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0048/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 49. Discriminant and Distinct Real Roots

Over the reals, (-1)x²+(7)x+(8)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=81; distinct real roots=2

D=81; distinct real roots=2. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0049/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 50. Discriminant and Distinct Real Roots

Over the reals, (1)x²+(16)x+(65)=0 has a≠0. Find the discriminant D and the number of distinct real roots.

View answer and explanationD=-4; distinct real roots=0

D=-4; distinct real roots=0. Compute D=b²−4ac; negative, zero, and positive D give zero, one, and two distinct real roots respectively. A repeated root is counted once.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0050/)

Quadratic equations over the reals · qf-quadratic-discriminant · candidate





## 51. Exact Substitution of a Rational Root Candidate

Over the reals, (2)x²+(16)x+(24)=0 has a≠0. Is the reduced rational candidate x=-6/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0051/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 52. Exact Substitution of a Rational Root Candidate

Over the reals, (-4)x²+(-26)x+(-22)=0 has a≠0. Is the reduced rational candidate x=-5/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 8; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0052/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 53. Exact Substitution of a Rational Root Candidate

Over the reals, (6)x²+(30)x+(0)=0 has a≠0. Is the reduced rational candidate x=-5/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0053/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 54. Exact Substitution of a Rational Root Candidate

Over the reals, (-2)x²+(-7)x+(9)=0 has a≠0. Is the reduced rational candidate x=-4/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 5; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0054/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 55. Exact Substitution of a Rational Root Candidate

Over the reals, (4)x²+(8)x+(-32)=0 has a≠0. Is the reduced rational candidate x=-4/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0055/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 56. Exact Substitution of a Rational Root Candidate

Over the reals, (-6)x²+(-33)x+(-42)=0 has a≠0. Is the reduced rational candidate x=-3/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 3; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0056/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 57. Exact Substitution of a Rational Root Candidate

Over the reals, (2)x²+(8)x+(6)=0 has a≠0. Is the reduced rational candidate x=-3/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0057/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 58. Exact Substitution of a Rational Root Candidate

Over the reals, (-4)x²+(-10)x+(0)=0 has a≠0. Is the reduced rational candidate x=-2/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 4; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0058/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 59. Exact Substitution of a Rational Root Candidate

Over the reals, (6)x²+(6)x+(-12)=0 has a≠0. Is the reduced rational candidate x=-2/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0059/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 60. Exact Substitution of a Rational Root Candidate

Over the reals, (-2)x²+(1)x+(6)=0 has a≠0. Is the reduced rational candidate x=-1/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 3; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0060/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 61. Exact Substitution of a Rational Root Candidate

Over the reals, (4)x²+(12)x+(8)=0 has a≠0. Is the reduced rational candidate x=-1/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0061/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 62. Exact Substitution of a Rational Root Candidate

Over the reals, (-6)x²+(-9)x+(-3)=0 has a≠0. Is the reduced rational candidate x=0/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -3; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0062/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 63. Exact Substitution of a Rational Root Candidate

Over the reals, (2)x²+(0)x+(0)=0 has a≠0. Is the reduced rational candidate x=0/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0063/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 64. Exact Substitution of a Rational Root Candidate

Over the reals, (-4)x²+(6)x+(-2)=0 has a≠0. Is the reduced rational candidate x=1/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0064/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 65. Exact Substitution of a Rational Root Candidate

Over the reals, (6)x²+(-18)x+(12)=0 has a≠0. Is the reduced rational candidate x=1/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0065/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 66. Exact Substitution of a Rational Root Candidate

Over the reals, (-2)x²+(-1)x+(6)=0 has a≠0. Is the reduced rational candidate x=2/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -4; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0066/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 67. Exact Substitution of a Rational Root Candidate

Over the reals, (4)x²+(-4)x+(-8)=0 has a≠0. Is the reduced rational candidate x=2/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0067/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 68. Exact Substitution of a Rational Root Candidate

Over the reals, (-6)x²+(15)x+(0)=0 has a≠0. Is the reduced rational candidate x=3/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -9; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0068/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 69. Exact Substitution of a Rational Root Candidate

Over the reals, (2)x²+(-8)x+(6)=0 has a≠0. Is the reduced rational candidate x=3/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0069/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 70. Exact Substitution of a Rational Root Candidate

Over the reals, (-4)x²+(22)x+(-28)=0 has a≠0. Is the reduced rational candidate x=4/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -4; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0070/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 71. Exact Substitution of a Rational Root Candidate

Over the reals, (6)x²+(-12)x+(-48)=0 has a≠0. Is the reduced rational candidate x=4/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0071/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 72. Exact Substitution of a Rational Root Candidate

Over the reals, (-2)x²+(7)x+(9)=0 has a≠0. Is the reduced rational candidate x=5/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -6; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0072/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 73. Exact Substitution of a Rational Root Candidate

Over the reals, (4)x²+(-20)x+(0)=0 has a≠0. Is the reduced rational candidate x=5/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0073/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 74. Exact Substitution of a Rational Root Candidate

Over the reals, (-6)x²+(39)x+(-33)=0 has a≠0. Is the reduced rational candidate x=6/1, with positive denominator, a solution?

View answer and explanationNo

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is -15; comparison with zero gives No.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0074/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 75. Exact Substitution of a Rational Root Candidate

Over the reals, (2)x²+(-16)x+(24)=0 has a≠0. Is the reduced rational candidate x=6/1, with positive denominator, a solution?

View answer and explanationYes

Since q=1>0, clear denominators by multiplying the substituted expression by q². Its numerator is 0; comparison with zero gives Yes.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0075/)

Quadratic equations over the reals · qf-quadratic-root-verification · candidate





## 76. Exact Root Set of a Quadratic Equation

Over the reals, (1)x²+(0)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{0/1}

D=0. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0076/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 77. Exact Root Set of a Quadratic Equation

Over the reals, (-1)x²+(2)x+(-1)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{1/1}

D=0. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0077/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 78. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(32)x+(10)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-5/1, -1/3}

D=784. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-5/1, -1/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0078/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 79. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(-27)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-9/2, 0/1}

D=729. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-9/2, 0/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0079/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 80. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(22)x+(-8)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-4/1, 1/3}

D=676. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-4/1, 1/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0080/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 81. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(-17)x+(14)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-7/2, 2/3}

D=625. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-7/2, 2/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0081/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 82. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(12)x+(-18)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-3/1, 1/1}

D=576. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-3/1, 1/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0082/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 83. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(-21)x+(-15)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-5/2, -1/1}

D=81. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-5/2, -1/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0083/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 84. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(16)x+(8)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-2/1, -2/3}

D=64. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-2/1, -2/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0084/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 85. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(-11)x+(-3)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-3/2, -1/3}

D=49. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-3/2, -1/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0085/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 86. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(6)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-1/1, 0/1}

D=36. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/1, 0/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0086/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 87. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(-1)x+(1)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-1/2, 1/3}

D=25. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/2, 1/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0087/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 88. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(-4)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{0/1, 2/3}

D=16. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1, 2/3}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0088/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 89. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(9)x+(-3)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{1/2, 1/1}

D=9. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/2, 1/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0089/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 90. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(0)x+(-6)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-1/1, 1/1}

D=144. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/1, 1/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0090/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 91. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(5)x+(6)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-2/3, 3/2}

D=169. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-2/3, 3/2}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0091/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 92. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(-10)x+(-4)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-1/3, 2/1}

D=196. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/3, 2/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0092/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 93. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(15)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{0/1, 5/2}

D=225. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1, 5/2}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0093/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 94. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(-20)x+(6)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{1/3, 3/1}

D=256. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/3, 3/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0094/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 95. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(25)x+(-14)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{2/3, 7/2}

D=289. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {2/3, 7/2}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0095/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 96. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(-30)x+(24)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{1/1, 4/1}

D=324. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {1/1, 4/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0096/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 97. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(21)x+(27)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-1/1, 9/2}

D=1089. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/1, 9/2}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0097/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 98. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(-26)x+(-20)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-2/3, 5/1}

D=1156. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-2/3, 5/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0098/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 99. Exact Root Set of a Quadratic Equation

Over the reals, (-6)x²+(31)x+(11)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{-1/3, 11/2}

D=1225. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {-1/3, 11/2}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0099/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate





## 100. Exact Root Set of a Quadratic Equation

Over the reals, (6)x²+(-36)x+(0)=0 has a≠0 and a nonnegative perfect-square discriminant. List the exact root set in increasing order as reduced fractions with positive denominators, writing each repeated value once.

View answer and explanation{0/1, 6/1}

D=1296. Apply (-b±√D)/(2a), reduce the rational roots, sort them, and remove duplicate values to obtain {0/1, 6/1}. The comparison is exact.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r04-quadratic-equation-0100/)

Quadratic equations over the reals · qf-quadratic-exact-roots · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-r04-quadratic-equation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-r04-quadratic-equation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-r04-quadratic-equation-foundations/validation-report.json)