It Always Works

The quadratic x^2 + 3x + 1 = 0 cannot be factored with whole numbers. No pair of whole numbers has product 1 and sum 3. The quadratic formula still works. It gives x = (-3 ± √5) / 2.

The discriminant is 5, which is not a perfect square. So the square root stays as √5, as in the square root chapter. The two solutions are exact numbers, and the formula found them even though factoring could not.

The quadratic formula works even when a quadratic cannot be factored with whole numbers. When the coefficients are whole numbers, a discriminant that is a perfect square gives solutions that are whole numbers or fractions. A positive discriminant that is not a perfect square gives solutions with a square root sign. Completing the square is another way to find the same solutions.