Quadratic Equation Solver

Enter the coefficients a, b and c of ax² + bx + c = 0 to find the roots, discriminant and vertex.

How to use the Quadratic Equation Solver

  1. Write your equation in the form ax² + bx + c = 0.
  2. Enter the values of a, b and c (use a minus sign for negative numbers).
  3. Read the roots, the discriminant and the vertex of the parabola.

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

For x² − 3x + 2 = 0, a = 1, b = −3 and c = 2. The discriminant is (−3)² − 4 × 1 × 2 = 1, so x = (3 ± 1) ÷ 2, giving x = 1 or x = 2.

What the discriminant tells you

  • b² − 4ac > 0: two different real roots; the parabola crosses the x-axis twice.
  • b² − 4ac = 0: one repeated real root; the parabola touches the x-axis at its vertex.
  • b² − 4ac < 0: two complex roots; the parabola does not reach the x-axis.

Vertex

The turning point of the parabola is at x = −b ÷ 2a, and its y-value is c − b² ÷ 4a. The parabola opens upward when a is positive and downward when a is negative.

Frequently asked questions

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ 2a, which gives the solutions of ax² + bx + c = 0.

What if the discriminant is negative?

The equation has no real solutions. It has two complex roots of the form p ± qi.

Why can a not be zero?

With a = 0 the x² term disappears and the equation becomes linear (bx + c = 0), which has a single solution x = −c ÷ b.

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