Can the quadratic formula solve every quadratic?

Yes — every quadratic equation, including ones with complex roots.

Quadratic Formula

Solves any quadratic equation ax² + bx + c = 0.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Variables

a, b, c: Coefficients of ax² + bx + c = 0 (a ≠ 0)
x: The solutions (roots)

Explanation

The quadratic formula comes from completing the square on $a x^{2} + b x + c = 0$ . The expression under the square root, $b^{2} - 4 a c$ , is the discriminant: positive gives two real roots, zero gives one repeated root, and negative gives two complex roots.

Example

Solve $x^{2} - 3 x - 10 = 0$ : $a = 1, b = - 3, c = - 10$ . Then $Δ = 9 + 40 = 49$ , so $x = \frac{3 \pm 7}{2}$ , giving $x = 5$ or $x = - 2$ .

Try it: Quadratic Formula Calculator

Solve ax² + bx + c = 0 with the quadratic formula. Shows the discriminant, exact roots, decimal values and factoring.

Open calculator

Frequently asked questions

Related formulas

Vertex of a Parabola

Finds the turning point (vertex) of y = ax² + bx + c.

Difference of Squares

Factors the difference of two perfect squares.