QESA

Math & Statistics

Quadratic Equation Solver and Verification Calculator

Solve a real quadratic equation and verify the discriminant, both branches, root sum, root product, direct substitution, and comparison gap.

Coefficient aCoefficient bCoefficient cEntered x for verificationEntered root comparison

Discriminant-

Square root of nonnegative discriminant-

Two times a-

Positive-branch real root-

Negative-branch real root-

Calculated root sum-

Viete root-sum reference-

Calculated root product-

Viete root-product reference-

Polynomial value at entered x-

Root one minus entered comparison-

Decision view

Quadratic parabola, roots, and verification point

Quadratic parabola, roots, and verification pointThe live function curve shows x-axis intersections, symmetry axis, vertex, and the entered substitution point.

Exact scenario comparisonCoefficient c changes while all other entered assumptions remain constant.

Coefficient c	Discriminant	Square root of nonnegative discriminant	Two times a	Positive-branch real root	Negative-branch real root	Calculated root sum	Viete root-sum reference	Calculated root product	Viete root-product reference	Polynomial value at entered x	Root one minus entered comparison

How to use Quadratic Equation Solver and Verification Calculator

Enter nonzero coefficient a and coefficients b and c.
Enter an x value for direct verification.
Enter a root comparison and decimal precision reference.
Inspect the parabola, axis, roots, vertex, and verification point.

Calculator guide

Understanding Quadratic Equation Solver and Verification Calculator

A quadratic solution should expose the discriminant before producing roots and should verify those roots independently. Viete's sum and product provide algebraic checks, while direct substitution confirms that a selected x makes the polynomial zero.

Discriminant first It classifies real roots.

Two branches Plus and minus are both required.

Independent checks Sum, product, and substitution agree.

Graph explains Roots are x-axis intersections.

Calculation method

How the calculation works

Solve the quadratic equation from its discriminant and two real branches, then verify the roots independently through their sum, product, and direct polynomial substitution. Calculate the discriminant, apply the positive and negative square-root branches over twice a, then compare the roots with Viete's identities and a direct polynomial evaluation.

Detailed calculation process

Solve and verify both real quadratic roots

The default polynomial is x^2 - 5x + 6, with verification x = 2 and comparison root = 3.

General formula: D = b^2-4acx_1 = (-b+sqrt(D))/(2a)x_2 = (-b-sqrt(D))/(2a)x_1+x_2 = -b/ax_1x_2 = c/af(x_v) = ax_v^2+bx_v+c The discriminant determines real-root availability. The plus and minus branches create two roots, and Viete's identities plus direct substitution verify the results without reusing the same branch arithmetic.

What each symbol means

a, b, c Quadratic coefficients in ax^2+bx+c.

D Discriminant.

sqrt(D) Nonnegative real square-root term.

x_1, x_2 Positive-branch and negative-branch real roots.

-b/a Viete root-sum reference.

c/a Viete root-product reference.

x_v, f(x_v) Verification input and polynomial value.

Worked substitution with the default inputs

1. Calculate the discriminant D = (-5)^2-4(1)(6) = 25-24 = 1 A positive discriminant means two distinct real roots.

2. Calculate the denominator and square root sqrt(D) = 12a = 2(1) = 2 Both root branches share the same denominator.

3. Solve both branches x_1 = [5+1]/2 = 3x_2 = [5-1]/2 = 2 The sign before the square root distinguishes the two solutions.

4. Verify with Viete x_1+x_2 = 3+2 = 5 = -b/ax_1x_2 = 3(2) = 6 = c/a Both independent coefficient identities reconcile exactly.

5. Verify by substitution f(2) = 1(2^2)-5(2)+6 = 4-10+6 = 0x_1-3 = 0 The entered x is a root and the positive branch equals the comparison root.

The default polynomial has roots 3 and 2; their sum, product, and direct substitution all reconcile exactly.

Mathematical function view

Plot the parabola and both coordinate intersections

The live coordinate plot shows the parabola, x-axis roots, symmetry axis, vertex, and the entered verification point, updating with every coefficient.

Parabola Function ax squared plus bx plus c.

Root markers X-axis intersections.

Symmetry axis Vertical line through the vertex.

Verification point Entered x and polynomial value.

Worked situations

Practical examples

The discriminant is 1, so there are two real roots.
The roots are 3 and 2.
Their sum is 5 and product is 6, matching the coefficients.

Better inputs

Useful tips

Check that coefficient a is nonzero.
Use the discriminant before interpreting real roots.
Verify with substitution when exact arithmetic is important.

Before relying on the result

Limitations and common mistakes

Negative discriminants require complex arithmetic and are not represented by this real-root model.
If a is zero, the equation is linear rather than quadratic.
Very large or nearly cancelling coefficients may require numerically stable methods.

Reference

Key terms

Discriminant: b squared minus 4ac, determining real-root structure.
Root: An x value where the polynomial equals zero.
Vertex: Turning point of the parabola.
Viete identities: Coefficient relationships for root sum and product.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why are there two roots?

The positive discriminant creates distinct plus and minus square-root branches.

What happens when the discriminant is zero?

Both branches meet at one repeated real root.

Why check sum and product?

They independently compare the calculated roots with the original coefficients.

Does this page solve complex roots?

No. The current model is explicitly limited to real-valued branches.