The Formula, Stated

The quadratic formula solves any equation of the form $ax^{2}+bx+c = 0$, as long as $a$ is not zero. The formula is $x = \frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$. It comes from completing the square. If the number under the square root is negative, the equation has no real solutions.

Here is the formula on $x^{2}+5x+6 = 0$, where $a = 1$, $b = 5$ and $c = 6$. Substitute the values: $x = \frac{-5\pm \sqrt{5^{2}-4\cdot 1\cdot 6}}{2\cdot 1}$. Work out the part under the root: $25-24 = 1$, and $\sqrt{1} = 1$. So $x = \frac{-5\pm 1}{2}$. The plus case gives -2 and the minus case gives -3. These are the same two solutions that factoring finds.

To use the quadratic formula, first substitute $a$, $b$ and $c$. Then work out the number under the square root and take its root. Finally, work out the plus case and the minus case separately. In this lesson, use the formula rather than factoring.