Quadratic Equation

The quadratic formula x = -b ± √b^2 - 4ac2a solves ax^2 + bx + c = 0.

The idea

Theorem (The quadratic formula).

If $a \neq 0$, then every solution of $ax^{2} + bx + c = 0$ is given by $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.$

To apply it, match the equation against $ax^{2} + bx + c = 0$ and read off the three coefficients with their signs. Every part of the formula is built from these three numbers, so one dropped minus sign — most often in the leading $-b$ — spoils the result.

The expression under the square root, $D = b^{2} - 4ac$, is the discriminant, and its sign settles how many real solutions there are. $D > 0$ gives two. $D = 0$ gives exactly one, since the $\pm$ changes nothing. $D < 0$ gives none, since no real number squares to a negative.

The $\pm$ places the two solutions at the same distance $\frac{\sqrt{b^{2} - 4ac}}{2a}$ on either side of $\frac{-b}{2a}$, which is how one formula delivers both at once.

Ways to work on it

• Walkthrough. Solve a quadratic equation with the quadratic formula, one piece at a time.
• Proof. Derive the quadratic formula from ax^2+bx+c=0 by completing the square.
• Practice. Compute the discriminant, then both roots of a random quadratic.
• Hardest. One question: the larger root of a quadratic with irrational roots and leading coefficient ≠ 1.

Not sure where to start? Take the ten-question placement test.