Skip to content
Calculator Tools
Algebra Calculators

Quadratic Equation Calculator

Solve ax² + bx + c = 0 with the quadratic formula: real or complex roots, discriminant, vertex and factored form.

Enter values

Results update as you type.

Result

Roots

-2, 5

Discriminant (b² − 4ac)
49
Nature of roots
Two distinct real roots
Vertex
(1.5, -12.25)
Factored form
(x + 2)(x - 5)
Step-by-stepShow
  1. D = (-3)² − 4(1)(-10) = 49
  2. x = (−(-3) ± √49) ÷ (2 × 1)
  3. x = -2 or 5

About the Quadratic Equation Calculator

Solve ax² + bx + c = 0 with the quadratic formula: real or complex roots, discriminant, vertex and factored form.

Also known as: quadratic formula calculator, quadratic solver, solve quadratic, roots of quadratic, discriminant calculator.

How to use this calculator

  1. Enter a, b, c.
  2. The result updates instantly as you type — press Calculate (or Enter) at any time.
  3. Open “Step-by-step” to see how the answer was worked out.
  4. Use Copy result or Share to save the answer or send a link with your inputs.

Formula

x = (−b ± √(b² − 4ac)) / 2a
a, b, c
= coefficients of ax² + bx + c = 0
D
= discriminant b² − 4ac

Worked example

Using the values pre-filled in the calculator above:

  • a = 1
  • b = -3
  • c = -10

the calculator returns:

  • Roots: -2, 5
  • Discriminant (b² − 4ac): 49
  • Nature of roots: Two distinct real roots
  • Vertex: (1.5, -12.25)

Working

  1. D = (-3)² − 4(1)(-10) = 49
  2. x = (−(-3) ± √49) ÷ (2 × 1)
  3. x = -2 or 5

Frequently asked questions

What does the discriminant tell you?

If b² − 4ac > 0 there are two real roots, if it equals 0 there is one repeated root, and if it is negative the roots are complex conjugates.

How do I find the vertex of a parabola?

The vertex is at x = −b/2a. Substitute that x back into the equation to get the y-coordinate.