Quadratic Formula Calculator With Steps (Roots, Vertex & Discriminant)

Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula. Enter a, b and c to get both roots (real or complex), the discriminant, the vertex and the axis of symmetry, with the steps shown. Example: x² − 5x + 6 = 0 has a = 1, b = −5, c = 6, a discriminant of 1, and roots x = 2 and x = 3.

How to use the Quadratic Formula Calculator

1. Write your equation in standard form ax² + bx + c = 0 (move every term to one side).
2. Type the coefficients a, b and c. Use a negative sign where needed, for example b = -5.
3. Press Calculate to see the roots, discriminant, vertex and axis of symmetry.
4. Open Show steps to see the substitution into the quadratic formula.

Formula and method

x = (−b ± √(b² − 4ac)) / 2a

The expression under the root, D = b² − 4ac, is the discriminant. If D > 0 there are two real roots, if D = 0 there is one repeated root, and if D < 0 there are two complex roots. The vertex of the parabola is at x = −b / 2a.

Example: 2x² + 3x − 2 = 0. D = 9 + 16 = 25, so x = (−3 ± 5) / 4, giving x = 0.5 and x = −2.

Frequently asked questions

What if the discriminant is negative?

The equation has no real solutions, but it has two complex solutions of the form p ± qi. The parabola does not cross the x-axis.

Can a equal zero?

No. If a = 0 the equation is linear (bx + c = 0), not quadratic. The calculator tells you the linear solution instead.

Why are there two answers?

A parabola can cross the x-axis at two points. The ± in the formula gives one root with the plus sign and one with the minus sign.

Does the quadratic formula always work?

Yes, for any quadratic with real coefficients. Factoring or completing the square can be faster for simple equations, but the formula never fails.

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