Question

Solve (x² + 3x)² - (x² + 3x) -6 = 0.

Answer 1

(x² + 3x)² - (x² + 3x) -6 = 0
Putting x² + 3x = y, the given equation becomes
y² - y - 6 = 0
⇒ y² - 3y + 2y - 6 = 0
⇒ y(y - 3) + 2(y - 3) = 0
⇒ (y - 3) (y + 2) = 0
⇒ y - 3 = 0 or y + 2 = 0
⇒ y = 3 or y = -2
But x² + 3x = y
x² + 3x = 3
⇒ x² + 3x - 3 = 0
Here a = 1, b = 3, c = -3
Then x = ${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$
x = ${\displaystyle \frac{-3\pm \sqrt{9+12}}{2}}$
x = ${\displaystyle \frac{-3\pm \sqrt{21}}{2}}$
or
x² + 3x = -2
x² + 3x + 2 = 0
x² + 2x + x + 2 = 0
x (x + 2) +1(x + 2) = 0
x + 2 = 0 or x + 1 = 0
x = -2 or x = -1
Hence, roots are ${\displaystyle \frac{-3\pm \sqrt{21}}{2},-2,-1}$.