Quadratic Formula Calculator

The quadratic formula

The part under the square root, D = b² − 4ac, is the discriminant. It tells you what kind of roots to expect:

• D > 0: two different real roots; the parabola crosses the x-axis twice. If D is a perfect square, the roots are whole numbers or fractions.
• D = 0: one repeated root; the parabola touches the x-axis at its vertex.
• D < 0: no real roots; the parabola misses the x-axis, and the roots are complex numbers.

Worked example: 2x² + 3x − 2 = 0

1. a = 2, b = 3, c = −2.
2. D = 3² − 4 × 2 × (−2) = 9 + 16 = 25, a perfect square.
3. x = (−3 ± 5) ÷ 4, so x = 2 ÷ 4 = 1/2 or x = −8 ÷ 4 = −2.
4. Factored: 2x² + 3x − 2 = (x + 2)(2x − 1).
5. Check: the roots add to −3/2 = −b/a and multiply to −1 = c/a.

Exact answers

When D isn’t a perfect square the roots are irrational, so this calculator writes them exactly, with the square root simplified: for x² − 2x − 1 = 0, x = 1 ± √2 (about −0.414 and 2.414). Decimals and fractions in the equation are cleared first by multiplying every term by the same number, which doesn’t change the roots.

Vertex and vertex form

The parabola’s turning point is at x = −b ÷ 2a, halfway between the roots. Writing the equation as a(x − h)² + k, called vertex form or completing the square, shows the vertex (h, k) directly.