Quadratic equations over the reals

Mathematical statement

Over the reals, let q(x)=ax²+bx+c with a,b,c∈R and a≠0. Then q(x)=0 is a quadratic equation in one variable. Put D=b²−4ac: there are no real roots when D<0, one distinct real root when D=0, and two distinct real roots when D>0. For D≥0, the roots are (-b±√D)/(2a).

Representative formula (exact TeX)

q(x)=0

Assumptions

• The domain is R and a≠0; an equation with a=0 is not quadratic.
• Use the principal square root only when D≥0. List each root-set value once; multiplicity is distinct from the number of distinct roots.

Symbol scope

• q — quadratic polynomial
• x — real unknown

Intuition, definition, and misconceptions

First check a≠0, then compute the discriminant. Its sign classifies real roots only: no real roots for D<0 does not mean no solutions in every number system.

When a≠0, completing the square gives q(x)=0 ⇔ (2ax+b)²=D. For D≥0 this yields the root formula. At D=0 the two formula branches coincide, and the root set contains one value.

The exercises use integer coefficients, substitution of rational root candidates after clearing denominators, and exact root sets for nonnegative perfect-square discriminants. They do not compare roots by floating approximations or claim a completed formal proof for every real square root.

Common errors include omitting a≠0, counting two distinct roots at D=0, or confusing absence of real roots with absence of complex roots.

Prerequisites

• Equation — First specify the domain of the unknown and the substitution condition for a solution.
• Natural-number powers and principal square roots — The square and the principal square root for D≥0 support the root formula.
• Univariate polynomials over the rationals — Integer-coefficient exercises are an exact subclass of real quadratic polynomials.

Verification boundary

Discriminants and rational roots of specified equations can be recomputed exactly; the general root formula and real-root classification have no attached formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind the nonzero quadratic coefficient over R, the completing-the-square equivalence, discriminant classification, and exact root sets.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.