Question

What are the solutions of the equation (x + 2)2 – 2(x + 2) – 15 = 0? Use u substitution to solve. (Will get medal)

Answer 1

set u=(x+2)
\[u ^{2}-2u-15=0\]
let's concentrate on just the left side
We can solve this by factoring:
x² - 2x - 15 = 0
x² - 5x + 3x - 15 = 0
x(x - 5) + 3(x - 5) = 0
(x - 5)(x + 3) = 0
x - 5 = 0 → x = 5
x + 3 = 0 → x = -3

We can also solve this, by completing the square:
x² - 2x - 15 = 0
x² - 2x = 15
x² - 2x + (-1)² = 15 + (-1)²
x² - 2x + 1 = 15 + 1
x² - 2x + 1 = 16
(x - 1)(x - 1) = 16
√(x - 1)² = ±16
x - 1 = ±4
x - 1 = 4 → x = 5
x - 1 = -4 → x = -3

did the rest in x in case you didn't notice
but u=2 and u=-3

and for the last part
u=x+2 so to find the real x
x=2-u plug in your u and you'll find your two answers.