Question

Solve: x(x + 1) (x + 3) (x + 4) = 180.

Answer 1

Given equation
x(x + 1) (x + 3) (x + 4) = 180
⇒ [(x + 0) (x + 4)] [(x + 1) (x + 3)] = 180
⇒ (x² + 4x) (x² + 4x + 3) - 180 = 0
Put x² + 4x = y,
then it becomes y(y + 3) - 180 = 0
⇒ y² + 3y - 180 = 0
⇒ y² + 15y - 12y - 180 = 0
⇒ y(y + 15) - 12(y + 15) = 0
⇒ (y + 15) (y - 12) = 0
⇒ y = 15 = 0 or y  = 12
But x² + 4x = y
Then x² + 4x = -15
x² + 4x + 15 = 0
or
x² + 4x = 12
⇒ x² + 4x - 12 = 0
x² + ax + 15 = 0 gives x = `(-4 ± sqrt((4)^2 - 4 xx 15))/(2)`
= `(-4 ± sqrt(16 - 60))/(2)`
= `(-4 ± sqrt(-34))/(2)`
∴ Roots of the equation are imaginary hence not acceptable.
or
x² + 4x - 12 = 0
⇒ x² + 6x - 2x - 12 = 0
⇒ x (x + 6) -2 (x + 6) = 0
⇒ (x + 6) (x - 2) = 0
⇒ x + 6 = 0 or x - 2 = 0
⇒ x = -6 or x = 2.