Quadratic Formula Calculator

Solve quadratic equations ax²+bx+c=0. Find roots, vertex, axis of symmetry, and discriminant.

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ax² + bx + c = 0

a (coefficient of x²)

b (coefficient of x)

c (constant)

x = (−b ± √(b²−4ac)) / 2a

Discriminant (Δ)

Enter coefficients to solve

Root x₁

Root x₂

Vertex (h, k)

Axis of Symmetry

Y-intercept

Opens

Property	Value

Solve any quadratic equation ax² + bx + c = 0 for its roots, discriminant, vertex, axis of symmetry, and y-intercept.

How It Works

How Quadratic Formula Calculator Works

The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, solves for both roots at once. The discriminant, Δ = b² − 4ac, determines what kind of roots exist without needing to compute them: a positive discriminant means two distinct real roots, zero means exactly one repeated real root, and a negative discriminant means two complex (imaginary) roots.

The vertex — the parabola's highest or lowest point — is found by completing the square: its x-coordinate is h = −b/2a, and substituting that back into the original equation gives the y-coordinate k. The vertical line x = h is the parabola's axis of symmetry, since the curve is a mirror image on either side of it.

The sign of coefficient a determines which way the parabola opens (upward if positive, downward if negative), and Vieta's formulas give the sum (−b/a) and product (c/a) of the two roots directly from the coefficients, without needing to solve for the individual roots first.

Worked Example

See It In Action

For x² − 5x + 6 = 0 (a=1, b=−5, c=6): discriminant Δ = (−5)² − 4(1)(6) = 25 − 24 = 1 (positive, so two real roots). Roots = (5 ± 1)/2, giving x₁ = 3 and x₂ = 2 — which checks out, since (x−2)(x−3) expands back to x²−5x+6. The vertex is at (2.5, −0.25), the axis of symmetry is x=2.5, the y-intercept is (0, 6), and the parabola opens upward.

Real-World Use Cases

Who Uses Quadratic Formula Calculator and Why

Solving for the roots of a quadratic equation on an algebra assignment.
Finding the vertex (maximum or minimum point) of a parabola modeling something like projectile motion.
Checking the discriminant's sign before graphing to know in advance how many real solutions to expect.
Verifying a factored answer using Vieta's formulas, comparing the sum and product of the roots to −b/a and c/a.

Common Mistakes

Mistakes to Avoid

Entering a=0, which turns the equation linear rather than quadratic — this calculator requires a nonzero a to apply the quadratic formula at all.
Interpreting a negative discriminant as 'no solution' rather than 'two complex roots' — the parabola simply never crosses the x-axis in that case, but complex solutions still exist.
Making a sign error on b when computing by hand and cross-checking — the formula uses −b, so a positive b becomes negative inside the calculation.

Pro Tips

Tips for Best Results

Check the discriminant's sign first as a quick preview of what kind of roots to expect before working through the full formula.
Use the vertex's x-coordinate (h = −b/2a) as a fast way to confirm the axis of symmetry before or after graphing.

Troubleshooting

Fixing Common Problems

The calculator won't compute a result. — Check that a is not zero — with a=0 the equation is linear, not quadratic, and the quadratic formula doesn't apply.

Glossary

Terms Explained

Discriminant: The value b² − 4ac, which determines whether a quadratic has two real roots (positive), one repeated real root (zero), or two complex roots (negative).

Vieta's formulas: Formulas giving the sum (−b/a) and product (c/a) of a quadratic's roots directly from its coefficients, without solving for the roots individually.

FAQ

Frequently Asked Questions

What does the discriminant tell me?

The discriminant (b²−4ac) predicts the nature of the roots before you even solve for them: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots involving imaginary numbers.

What happens if a=0?

If a=0, the equation is no longer quadratic — it becomes linear (bx+c=0) — so this calculator requires a nonzero value for a to apply the quadratic formula.

What are complex roots, and when do they appear?

Complex roots appear whenever the discriminant is negative, meaning the parabola never crosses the x-axis. They're expressed in the form (real part) ± (imaginary part)i, where i = √−1.

How is the vertex related to the roots?

When the equation has two real roots, the axis of symmetry (and therefore the vertex's x-coordinate) always lies exactly halfway between them, since a parabola is perfectly symmetric around its vertex.