Quadratic Equation Solver

Live discriminant nature gate for ax² + bx + c = 0, with a ± formula work peel, vertex/axis strip, and Vieta sum/product check for real or complex roots.

Educational algebra literacy only. Not a CAS, exam oracle, or engineering certification. When a = 0 the board falls back to a linear solve. Confirm method requirements with your course.

By Ahmet C. Toplutaş·Site owner & editor · Math tools

Last updated May 21, 2026 · Reviewed May 21, 2026 · Planning estimates only - not tax, medical, or investment advice

Discriminant sign decides root nature

Before memorizing the ± formula, read the discriminant. For ax² + bx + c = 0, D = b² − 4ac tells you whether the roots are two distinct reals, one repeated real, or a complex conjugate pair.

This page's framing is a nature gate, with a ± formula work peel, a vertex/axis strip, and a Vieta check. It is not a proportion cross-product twin, not a P vs C identity board, and not a refinance path twin.

Decision framing: “What does the sign of D say, and do the formula steps and Vieta check agree?”

Educational algebra literacy only. Not a computer algebra system, not an exam oracle, and not engineering certification. When your course requires factoring or completing the square, follow that method and use this board to verify. See our disclaimers.

± formula work peel

The quadratic formula is x = (−b ± √D) / (2a). The peel shows D, √|D|, the ± numerators when roots are real, and the 2a denominator so a sign error in a is visible.

When D < 0, the same structure yields complex conjugates: real part −b/(2a) and imaginary part ±√(−D)/(2a). The gate stays lit on D < 0 so you do not force a real square root of a negative number on a basic calculator.

If a = 0, the board drops to the linear solve x = −c/b instead of dividing by 2a.

Vertex and axis of symmetry

A true quadratic (a ≠ 0) is a parabola. The axis of symmetry is x = −b/(2a). The vertex is that x-value paired with f(−b/(2a)). The curve opens up when a > 0 and down when a < 0.

Vertex literacy connects algebra to graphing: the repeated-root case sits exactly on the axis, and complex roots mean the parabola never crosses the x-axis over the reals.

Vieta check

For a ≠ 0, the roots satisfy sum = −b/a and product = c/a. After the formula peel, read Vieta as a second opinion. Complex conjugates still obey the same sum and product over the complexes.

Core relations

Standard form, discriminant, formula, vertex, and Vieta in one place.

ax² + bx + c = 0 (a ≠ 0) D = b² − 4ac x = (−b ± √D) / (2a) Vertex: (−b/(2a), f(−b/(2a))) Sum = −b/a · Product = c/a

Example: 2x² + 5x − 3 = 0 → D = 49 → roots 1/2 and −3. Vieta: sum = −5/2, product = −3/2.

Where this hands off

Sibling boards for graphing helpers, geometry, and general numeric work.

Slope calculator (line literacy beside parabolas)
Triangle calculator
Scientific calculator
Proportion / permutations & combinations
Terms: math glossary

Worked scenarios

Scenario A: Two real. 2x² + 5x − 3 = 0 → D = 49 → roots 0.5 and −3. Nature gate on D > 0; Vieta sum −2.5, product −1.5.

Load Complex: x² + 2x + 5 = 0 → D = −16 → roots −1 ± 2i. Gate on D < 0; vertex at (−1, 4).

Scenario B: Repeated. x² − 6x + 9 = 0 → D = 0 → root 3 (double). Vertex sits on the x-axis at (3, 0).

Scenario C: Opens up. x² − 4x + 1 = 0 → two real roots near 0.268 and 3.732; axis at x = 2.

Scenario D: Linear fallback. Set a = 0, b = 2, c = −8 → x = 4. The board labels linear instead of forcing 2a in the denominator.

Common mistakes

Forgetting a ≠ 0 before calling the equation quadratic.
Taking √D on a calculator when D is negative (need complex form).
Sign errors on −b or on 2a in the denominator.
Skipping Vieta when a quick sum/product check would catch an arithmetic slip.
Mixing vertex x = −b/(2a) with a root when D ≠ 0 (they only coincide for the repeated case).

How to use this board

Enter a, b, and c (or load a scene preset).
Read the nature gate for the sign of D.
Confirm roots, then walk the ± work peel.
Check vertex/axis and Vieta. Share writes a, b, c; typing does not rewrite the address bar.

When not to stop at this sketch

Stop here for standard-form quadratics and discriminant literacy. Hand off for general scientific evaluation, triangle geometry, or slope of a line. For symbolic factoring proofs, follow your course method.

General calc: scientific calculator.
Line slope: slope calculator.
Glossary: quadratic equation.

Frequently asked questions

How does this quadratic equation solver work?

Enter coefficients a, b, and c for ax² + bx + c = 0. The board computes the discriminant D = b² − 4ac, gates root nature by the sign of D, peels the ± quadratic formula steps, and shows the vertex/axis plus a Vieta sum/product check.

What does the discriminant tell you?

D > 0 means two distinct real roots, D = 0 means one repeated real root, and D < 0 means a complex conjugate pair. The nature gate highlights which case your coefficients hit.

What is the quadratic formula peel?

It expands x = (−b ± √D) / (2a): discriminant value, √|D|, the ± numerators when real, and division by 2a. Complex cases show real and imaginary parts instead of two real numerators.

What are the vertex and axis of symmetry?

For a true quadratic (a ≠ 0), the axis is x = −b/(2a) and the vertex is (−b/(2a), f(−b/(2a))). The parabola opens up when a > 0 and down when a < 0.

What is the Vieta check?

For ax² + bx + c = 0 with a ≠ 0, the roots satisfy sum = −b/a and product = c/a. Use it to verify formula results, including complex conjugates.

What if a = 0?

Then the equation is linear (bx + c = 0) when b ≠ 0, or degenerate when b = 0. The board labels that case instead of forcing the quadratic formula.

Is this a computer algebra system or exam oracle?

No. It is educational algebra literacy for standard-form quadratics. Not engineering certification, not symbolic CAS, and not a substitute for your course method when factoring or completing the square is required.

Sources

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Discriminant sign decides root nature

Decision framing: “What does the sign of D say, and do the formula steps and Vieta check agree?”

± formula work peel

The quadratic formula is x = (−b ± √D) / (2a). The peel shows D, √|D|, the ± numerators when roots are real, and the 2a denominator so a sign error in a is visible.

If a = 0, the board drops to the linear solve x = −c/b instead of dividing by 2a.

Vertex and axis of symmetry

A true quadratic (a ≠ 0) is a parabola. The axis of symmetry is x = −b/(2a). The vertex is that x-value paired with f(−b/(2a)). The curve opens up when a > 0 and down when a < 0.

Vertex literacy connects algebra to graphing: the repeated-root case sits exactly on the axis, and complex roots mean the parabola never crosses the x-axis over the reals.

Core relations

Standard form, discriminant, formula, vertex, and Vieta in one place.

ax² + bx + c = 0 (a ≠ 0) D = b² − 4ac x = (−b ± √D) / (2a) Vertex: (−b/(2a), f(−b/(2a))) Sum = −b/a · Product = c/a

Example: 2x² + 5x − 3 = 0 → D = 49 → roots 1/2 and −3. Vieta: sum = −5/2, product = −3/2.

Worked scenarios

Scenario A: Two real. 2x² + 5x − 3 = 0 → D = 49 → roots 0.5 and −3. Nature gate on D > 0; Vieta sum −2.5, product −1.5.

Load Complex: x² + 2x + 5 = 0 → D = −16 → roots −1 ± 2i. Gate on D < 0; vertex at (−1, 4).

Scenario B: Repeated. x² − 6x + 9 = 0 → D = 0 → root 3 (double). Vertex sits on the x-axis at (3, 0).

Scenario C: Opens up. x² − 4x + 1 = 0 → two real roots near 0.268 and 3.732; axis at x = 2.

Scenario D: Linear fallback. Set a = 0, b = 2, c = −8 → x = 4. The board labels linear instead of forcing 2a in the denominator.

Common mistakes

Forgetting a ≠ 0 before calling the equation quadratic.
Taking √D on a calculator when D is negative (need complex form).
Sign errors on −b or on 2a in the denominator.
Skipping Vieta when a quick sum/product check would catch an arithmetic slip.
Mixing vertex x = −b/(2a) with a root when D ≠ 0 (they only coincide for the repeated case).

When not to stop at this sketch

General calc: scientific calculator.
Line slope: slope calculator.
Glossary: quadratic equation.

Frequently asked questions

How does this quadratic equation solver work?

What does the discriminant tell you?

D > 0 means two distinct real roots, D = 0 means one repeated real root, and D < 0 means a complex conjugate pair. The nature gate highlights which case your coefficients hit.

What is the quadratic formula peel?

It expands x = (−b ± √D) / (2a): discriminant value, √|D|, the ± numerators when real, and division by 2a. Complex cases show real and imaginary parts instead of two real numerators.

What are the vertex and axis of symmetry?

For a true quadratic (a ≠ 0), the axis is x = −b/(2a) and the vertex is (−b/(2a), f(−b/(2a))). The parabola opens up when a > 0 and down when a < 0.

What is the Vieta check?

For ax² + bx + c = 0 with a ≠ 0, the roots satisfy sum = −b/a and product = c/a. Use it to verify formula results, including complex conjugates.

What if a = 0?

Then the equation is linear (bx + c = 0) when b ≠ 0, or degenerate when b = 0. The board labels that case instead of forcing the quadratic formula.

Discriminant sign decides root nature

± formula work peel

Vertex and axis of symmetry

Vieta check

Core relations

Where this hands off

Worked scenarios

Common mistakes

How to use this board

When not to stop at this sketch

Frequently asked questions

How does this quadratic equation solver work?

What does the discriminant tell you?

What is the quadratic formula peel?

What are the vertex and axis of symmetry?

What is the Vieta check?

What if a = 0?

Is this a computer algebra system or exam oracle?

Sources

Discriminant sign decides root nature

± formula work peel

Vertex and axis of symmetry

Vieta check

Core relations

Where this hands off

Worked scenarios

Common mistakes

How to use this board

When not to stop at this sketch

Frequently asked questions

How does this quadratic equation solver work?

What does the discriminant tell you?

What is the quadratic formula peel?

What are the vertex and axis of symmetry?

What is the Vieta check?

What if a = 0?

Is this a computer algebra system or exam oracle?

Sources

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