Quadratic Equations and the Quadratic Formula

A quadratic equation in one unknown has a nonzero squared term after all terms are moved to one side and combined. In standard form, a is the coefficient of the squared term, b is the coefficient of the linear term, and c is the constant. A missing term has coefficient zero.

What the discriminant tells you about the roots

The discriminant is the expression under the square root in the quadratic formula. A positive value gives two distinct real roots. At zero the two signs give the same root, so it is a repeated root. A negative value gives no real roots. On the graph of the quadratic, those three cases correspond to crossing the horizontal axis twice, touching it once, or missing it.

Use the formula when factoring is not obvious

For the example below, identify a as 2, b as −3 and c as −1. Square b with its sign inside parentheses. The discriminant is 17, so the square root stays in the exact answers. The denominator applies to the whole numerator for each sign.

The two values have sum one and a half, and product negative one half. These match the sum and product obtained by comparing coefficients with the factored form.

Missing coefficients can make the equation easier

When the constant is zero, take out the common factor x and set each factor equal to zero. Dividing the whole equation by x would discard the root zero. When the linear term is absent, isolate the squared term and take both real square roots if the resulting value is positive.

If the squared term alone equals zero, zero is the only root. If isolating the square makes it equal to a negative number, there is no real solution. The quadratic formula still applies whenever a is nonzero.