Computing It

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

x = (-b ± √(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 ax^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: (-6)^2 - 4 × 1 × 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: (-6)^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 × 1 × (-4), which is 9 + 16, or 25.