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

The roots of the equation are - 1 and 9

Step-by-step explanation:

* Lets represent the general form of the completing

square ⇒ a (x - b) ² + c, were a, b, c are constant

* Now lets study the problem

∵ x² - 8x = 9 ⇒ arrange the terms

∴ x² - 8x - 9 = 0

* Lets equate left hand side by the general form of quadratic

∴ x² - 8x - 9 = a (x - b) ² + c ⇒ solve the bracket

∴ x² - 8x - 9 = a (x² - 2bx + b²) + c ⇒ open the bracket

∴ x² - 8x - 9 = ax² - 2abx + ab² + c

* Now lets make a comparison between the two sided

∵ x² = ax² ⇒ : x²

∴ 1 = a

∵ - 8x = - 2abx ⇒ : x

∴ - 8 = - 2ab ⇒ substitute the value of a

∴ - 8 = - 2 (1) b ⇒ : - 2

∴ 4 = b

∵ ab² + c = - 9 ⇒ substitute the values of a and b

∴ (1) (4²) + c = - 9

∴ 16 + c = - 9 ⇒ subtract 16 from both sides

∴ c = - 25

* Now lets write the completing square

∴ x² - 8x - 9 = (x - 4) ² - 25

∵ x² - 8x - 9 = 0

∴ (x - 4) ² - 25 = 0

* Add 25 to both sides

∴ (x - 4) ² = 25 ⇒ take √ for both sides

∴ x - 4 = ± 5

∴ x - 4 = 5 ⇒ add 4 to both sides

∴ x = 9

OR

x - 4 = - 5 ⇒ add 4 to both sides

∴ x = - 1

* The roots of the equation are - 1 and 9