Question

Solve the following by reducing them to quadratic equations:
z⁴ - 10z² + 9 = 0.

Answer 1

Given equation z⁴ - 10z² + 9 = 0
Putting z² = x, then given equation reduces to the form x² - 10x + 9 = 0
⇒ x² - 9x - x + 9 = 0
⇒ x(x - 9) -1(x - 9) = 0
⇒ (x - 9) (x - 1) = 0
⇒ x - 9 = 0 or x - 1 = 1
⇒ x = 9 or x = 1
But z² = x
∴ z² = 9
⇒ z = ±3
or
z² = 1
z = ±1
Hence, the required roots are ±3, ±1.