11 COMPLETING THE SQUARE IF WE TRY TO SOLVE this quadratic equation by factoring x² + 6x + 2 = 0 we cannot. Therefore, we use a technique called completing the square. This means to make the quadratic into a perfect square trinomial, i.e. the form a² + 2ab + b² = (a + b)². The technique is valid only when 1 is the coefficient of x². 1) Transpose the constant term to the right: x² + 6x = −2 2) Add a square number to both sides. Add the square of half the coefficient of x. In this case, add the square of 3: x² + 6x + 9 = −2 + 9 The left-hand side is now the perfect square of (x + 3). (x + 3)² = 7 3 is half of the coefficient 6. This equation has the form
That is, the solutions to x² + 6x + 2 = 0 are the conjugate pair, −3 + , −3 − . We can check this. The sum of those roots is −6, which is the negative of the coefficient of x. And the product of the roots is (−3)² − ()² = 9 − 7 = 2, which is the constant term. Thus both conditions on the roots are satisfied. These are the two roots of the quadratic. Problem. Solve this quadratic equation by completing the square. x² − 2x − 2 = 0 To see the solution, pass your mouse over the colored area.
Before considering the quadratic formula, note that half of any
The quadratic formula Theorem. If ax² + bx + c = 0, Theorem. then To prove this, we will complete the square. But to do that, the coefficient of x² must be 1. Therefore, we will divide both sides of the original equation by a: This is the quadratic formula. Next Topic: Synthetic division by x − a Please make a donation to keep TheMathPage online. Copyright © 2001-2007 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |