Question

Solve for x: (x² - 5x)² - 7(x² - 5x) + 6 = 0; x ∈ R.

Answer 1

Given equation
(x² - 5x)² - 7(x² - 5x) + 6 = 0
Put x² - 5x = y
∴ The given equation becomes
y² - 7y + 6 = 0
⇒ y² - 6y - y + 6 = 0
⇒ y(y - 6) -1(y - 6) = 0
⇒ y = 1, 6
But x² - 5x = y
∴ x² - 5x = 1
x² - 5x - 1 = 0
Here a = 1, b = -5, c = -1
∴ x = `(-b ± sqrt(b^2 - 4ac))/(2a)`

x = `(-(-5) ± sqrt(25 + 4))/(2)`

x = `(5 ± sqrt(29))/(2)`

x² - 5x = 6
⇒ x² - 5x - 6 = 0
⇒ x² - 6x + x - 6 = 0
⇒ x(x - 6) +1(x - 6) = 0
⇒ (x - 6) (x + 1) = 0
⇒ x = 6 or x = -1
Hence, the roots are -1, 6, `(5 ± sqrt(29))/(2)`.