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

because linear equations are a lot simpler than quadratics equations, and linear equations formula is a lot easier to memorize. that's why i think they're harder.