Find the roots of the quadratic equation x2 - 8x = 9 by completing the square. Show your work.

The answer is - 1 or 9

The quadratic equation is ax² + bx + c = 0. But, by completing the square we turn it into: a (x + d) ² + e = 0, where:

d = b/2a

e = c - b²/4a

Our quadratic equation is x² - 8x = 9, which is after rearrangement:

x² - 8x - 9 = 0

So, a = 1, b = - 8, c = - 9

Let's first calculate d and e:

d = b/2a = - 8 / (2 * 1) = - 8/2 = - 4

e = c - b²/4a = - 9 - (-8) ² / (4 * 1) = - 9 - 64/4 = - 9 - 16 = - 25

By completing the square we have:

a (x + d) ² + e = 0

1 (x + (-4)) ² + (-25) = 0

(x - 4) ² - 25 = 0

(x - 4) ² = 25

⇒ x - 4 = √25

Since √25 can be either - 5 or + 5, then:

x - 4 = - 5 or x - 4 = 5

x = - 5 + 4 or x = 5 + 4

x = - 1 or x = 9