Answer:
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
