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Algebra · Calculator

Quadratic Formula Calculator

Enter the coefficients of any equation in the form ax² + bx + c = 0. AI-math returns the discriminant, exact roots, vertex and factored form — with every step shown and a graphed parabola.

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ax2 + bx + c = 0
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+
x
+
= 0
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Solution

WORKING Step-by-step solution
Graph of the parabola

What is the quadratic formula?

A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b and c are numbers and a is not zero. The quadratic formula gives the solutions (the values of x where the curve crosses the x-axis) directly from those three coefficients:

x = −b ± √(b² − 4ac)2a
The ± sign means there are usually two solutions — one using + and one using −.

The expression under the square root, b² − 4ac, is called the discriminant. It tells you what kind of roots to expect before you even finish solving:

  • Discriminant > 0 → two distinct real roots (the parabola crosses the x-axis twice).
  • Discriminant = 0 → one repeated real root (the parabola just touches the x-axis).
  • Discriminant < 0 → two complex conjugate roots (the parabola never touches the x-axis).

How to use this calculator

  1. Type the coefficient of x² into the first box (this is a).
  2. Type the coefficient of x into the second box (this is b). Use a minus sign for negative terms.
  3. Type the constant into the third box (this is c).
  4. Press Solve step by step. You'll get the discriminant, both roots, the vertex, the factored form and a graph — with the full working underneath.

Tip

No equation handy? Tap one of the example chips above the Solve button to load a worked problem instantly.

Worked example: x² − 3x − 10 = 0

Here's exactly what the calculator does for the default equation, so you can follow the method by hand.

01
Identify the coefficients: a = 1, b = −3, c = −10.
02
Compute the discriminant: D = b² − 4ac = (−3)² − 4(1)(−10) = 9 + 40 = 49.
03
Since D = 49 > 0, expect two real roots. Take the square root: √49 = 7.
04
Apply the formula: x = (3 ± 7) / 2.
05
Split the ±: x = (3 + 7)/2 = 5 and x = (3 − 7)/2 = −2.
06
Factored form: (x − 5)(x + 2) = 0. The roots are x = 5 and x = −2.

Common mistakes to avoid

  • Dropping a negative sign. In b² − 4ac, a negative c makes −4ac positive — a frequent slip.
  • Forgetting the ± gives two answers. Unless the discriminant is zero, a quadratic has two roots.
  • Dividing only part of the numerator by 2a. The whole −b ± √D is divided by 2a.

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Frequently asked questions

Yes. When the discriminant is negative, the parabola never crosses the x-axis, and the calculator returns the two complex conjugate roots in the form p ± qi, with the full working.

If a is zero the equation is no longer quadratic — it's linear (bx + c = 0). The calculator detects this and solves it as a linear equation instead, x = −c / b.

When the roots are rational, the calculator displays the factored form such as (x − 5)(x + 2) = 0 alongside the roots, so you can check the factorisation directly.

Yes — it's completely free to use with no account required. For unlimited daily use across all calculators plus the AI tutor, an affordable plan is available.