Solving quadratic equations

Standard form

A quadratic equation is one that can be written in standard form

If \(a\) were \(0\), the equation would be linear. A quadratic equation has at most two real solutions, also called roots.

There are four main ways to solve one. The first three are shortcuts for special cases; the fourth always works.

Method 1: square roots

When there's no \(x\) term (\(b = 0\)), isolate \(x^2\) (or a squared binomial) and take square roots. Don't forget the \(\pm\).

If isolating gives \(x^2\) equal to a negative number, as in \(x^2 = -9\), there's no real solution.

Method 2: factoring

So if the quadratic factors, set each factor equal to zero.

Method 3: completing the square

Any quadratic can be rewritten as a squared binomial plus a number. With \(a = 1\), add \(\left(\frac{b}{2}\right)^2\) to both sides.

Completing the square on the general equation \(ax^2 + bx + c = 0\) produces the quadratic formula.

Method 4: the quadratic formula

The discriminant

The expression under the square root, \(b^2 - 4ac\), is the discriminant. Its sign tells you how many real solutions there are before you finish solving.

Practice