Quadratic equation solver

The formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

This solves $ax^2 + bx + c = 0$ for any $a \neq 0$.

Where it comes from

The method is called completing the square. Start with the equation and divide through by $a$:

$$x^2 + \frac{b}{a}x = -\frac{c}{a}$$

The left side is almost a perfect square. $(x + \frac{b}{2a})^2$ expands to $x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}$, so add $\frac{b^2}{4a^2}$ to both sides:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$

Take the square root of both sides, remembering it can be positive or negative, then subtract $\frac{b}{2a}$. That gives the quadratic formula.

The discriminant $\Delta = b^2 - 4ac$ is the part under the root, and it decides what kind of answer you get:

• $\Delta > 0$: two real roots. The parabola crosses the x-axis twice.
• $\Delta = 0$: one repeated root. The parabola just touches the axis.
• $\Delta < 0$: two complex roots. The parabola never reaches the axis.

The vertex, the parabola’s turning point, sits halfway between the roots, at $x = -b/2a$.

Worked example

$2x^2 - 3x - 5 = 0$, so $a = 2$, $b = -3$, $c = -5$:

$$\Delta = 9 + 40 = 49$$ $$x = \frac{3 \pm 7}{4} = 2.5 \ \text{or}\ -1$$

Watch out for

• Signs. $-b$ means flip the sign of $b$. With $b = -3$, $-b = 3$.
• Rounding errors on computers. When $b^2$ is much larger than $4ac$, subtracting nearly equal numbers loses precision. This calculator uses the rearranged form from Numerical Recipes to avoid it.

Sources

• Press, W. H. et al. Numerical Recipes, 3rd edition (2007), section 5.6 "Quadratic and Cubic Equations"

Last checked November 11, 2026