Computing It

In the quadratic formula one part decides what kind of answers you get: the expression under the square root.

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

That expression, ${b}^{2}-4ac$, has its own name: the discriminant. When a, b and c are real, its sign tells you, before you solve anything, how many solutions there are and whether they are real or complex.

To compute it, read a, b and c off the equation in the form $a{x}^{2}+bx+c = 0$, signs included. For ${x}^{2}-6x+5 = 0$, a is 1, b is -6 and c is 5. Put them in: ${\left(-6\right)}^{2}-4\cdot 1\cdot 5$. The square is 36 and the product is 20, so the discriminant is $36-20$, which is 16.

Keep b in brackets when you square it: ${\left(-6\right)}^{2}$ is 36, never -36, because the square of a real number is never negative. The minus that can bite is the one in front of 4ac. When c is negative it turns into a plus: for ${x}^{2}+3x-4 = 0$ the discriminant is ${3}^{2}-4\cdot 1\cdot \left(-4\right)$, which is 9 + 16, or 25.