Completing the Square
Students learn to solve quadratic equations by completing the square. For example, to solve the equation s^2 - 6s + 5 = 0, the first step is put the constant term on the opposite side of the equation as the terms that contain the variables, by subtracting 5 from both sides, to get s^2 - 6s ___ = - 5 ___. Next, the number that goes in each space comes from half the coefficient of the middle term squared, which in this case is half of -6, or -3, squared, which is +9. So a +9 goes in each space, to get s^2 - 6s + 9 = - 5 + 9. The trinomial on the left side of the equation then factors as (s - 3)(s - 3), or (s - 3)^2, and the right side of the equation simplifies to 4, so the problem now reads (s - 3)^2 = 4. Finally, take the square root of both sides to get s - 3 = plus or minus 2, so s - 3 = 2, or s - 3 = -2, and solving each equation from here, s = 5 or 1.