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

Answer:The roots of the equation are -1 and 9Step-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 = 9ORx - 4 = -5 ⇒ add 4 to both sides∴ x = -1* The roots of the equation are -1 and 9