Completing the Square

Question

Complete the square so that the expression of the equation y=x²+6x+15 involves a squared binomial.

Answer 1

y = x² + 6x + 15. Let's complete the equation to get the squared binomial (x + b)² = a² + 2bx + b²

In our case, 2b = 6 => b = 3 and so b² = 3² is added and subtracted: added to complete the binomial and then subtracted for retaining the same value:

y = x² + 6x + 15 = x² + 2 _* 3 _* x + 3² - 3² + 15 = (x + 3)² - 9 + 15 = (x + 3)² + 6

Answer: y = (x + 3)² + 6

Answer 2

Answer 3

Consider a quadratic in the form Ax² + Bx + C. To complete the square, we add the square of one-half of B, (B / 2)². Then, after factoring, our expression looks like this: (x + B/2)²

For y = x² + 6x + 15, this means we "add" (6 / 2)² = 9. We already have 15, so our expression now looks like this:

y = (x² + 6x + 9) + 6 [the 9 and the 6 add up to the original 15]

The square binomial is of the form (x + B/2)², which is (x + 3)²

Our final expression is thus y = (x + 3)² + 6

To check your work, you can expand the square binomial and add the 6 to get back to your original equation.

Hope this was helpful.