Quadratic Formula Calculator

The quadratic formula

Every equation ax² + bx + c = 0 with a ≠ 0 is solved by

x = (−b ± √(b² − 4ac)) / 2a

The expression under the root, Δ = b² − 4ac, is the discriminant. It tells you how many real solutions there are:

• Δ > 0 – two different real roots; the parabola crosses the x-axis twice
• Δ = 0 – one double root; the parabola touches the x-axis at its vertex
• Δ < 0 – no real roots, two complex ones; the parabola never reaches the x-axis

Worked example: 2x² − 7x + 3 = 0

a = 2, b = −7, c = 3. Δ = (−7)² − 4 · 2 · 3 = 49 − 24 = 25, and √25 = 5. x = (7 ± 5) / 4, so x₁ = 0.5 and x₂ = 3. The factored form is (2x − 1)(x − 3).

The vertex

The parabola turns at x = −b / 2a. For 2x² − 7x + 3 that is x = 1.75, and y = 2 · 1.75² − 7 · 1.75 + 3 = −3.125. Because a is positive, the parabola opens upwards and the vertex is its lowest point.

Complex roots

For x² + 2x + 5 = 0, Δ = 4 − 20 = −16. The root of a negative number is imaginary: √(−16) = 4i, so x = (−2 ± 4i) / 2 = −1 ± 2i.

Exact answers

When a, b and c are whole numbers, the calculator also gives the exact form with a simplified root, for example x² − 3x + 1 = 0 has the roots x = (3 ± √5) / 2 – about 0.381966 and 2.618034.

Frequently asked questions