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 $x = \frac{-3\pm \sqrt{5}}{2}$.

The discriminant is 5, which is not a perfect square. So the square root stays as $\sqrt{5}$, 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.