Quadratic Formula and the Discriminant

Solves ax² + bx + c = 0 with the discriminant D = b² − 4ac and the quadratic formula x = (−b ± √D) ÷ 2a.

A quadratic equation $ax^2 + bx + c = 0$ is solved by the quadratic formula. Supply the three coefficients and the real roots drop out.

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

The quantity under the square root, $D = b^2 - 4ac$ , is the discriminant, and its sign alone decides how many real solutions exist.

Example

Take $a = 1$ , $b = -5$ , $c = 6$ , that is $x^2 - 5x + 6 = 0$ . The discriminant is

D = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1

Positive, so there are two real roots:

x = \dfrac{5 \pm \sqrt{1}}{2} = \dfrac{5 \pm 1}{2}

giving $x_1 = 3$ and $x_2 = 2$ . The factorisation $x^2 - 5x + 6 = (x - 2)(x - 3)$ confirms it.

Watch the sign of $b$ . When $b$ is negative, $-b$ is positive: here $-(-5) = 5$ . Sign slips are far and away the most common mistake with this formula.
With $a = 0$ the squared term disappears, the equation stops being quadratic, and the denominator $2a$ would be zero. The calculator rejects it.
A negative discriminant means there is no real root, and the calculator says so rather than returning a complex value.
As a check, the two roots must add up to $-\dfrac{b}{a}$ and multiply to $\dfrac{c}{a}$ . Here $3 + 2 = 5$ and $3 \times 2 = 6$ , exactly as expected.