Irrational Answers

Completing the square works whether or not the answers are whole numbers, which is where it beats factoring. Take x^2 + 4x - 1 = 0. No two whole numbers multiply to -1 and add to 4, so it does not factor. Completing the square still gives (x + 2)^2 = 5.

5 is not a perfect square, so the roots stay as roots: x + 2 is √(5) or -√(5), and x is -2 + √(5) or -2 - √(5). These are exact. A decimal like 0.236 would only be close.

Sometimes the root can be simplified. x^2 - 2x - 7 = 0 becomes (x - 1)^2 = 8, and √(8) is 2√(2) because 8 is 4 times 2. The answers are 1 + 2√(2) and 1 - 2√(2).