Question

What are the solutions of the equation x^4 + 6x^2 + 5 = 0? Use u substitution to solve.

Answer 1

second option

Explanation:

Given

`x^(4)` + 6x² + 5 = 0

let u = x², then

u² + 6u + 5 = 0 ← in standard form

(u + 1)(u + 5) = 0 ← in factored form

Equate each factor to zero and solve for u

u + 1 = 0 ⇒ u = - 1

u + 5 = 0 ⇒ u = - 5

Change u back into terms of x, that is

x² = - 1 ( take the square root of both sides )

x = ±
`√(-1)` = ± i ( noting that
`√(-1)` = i ), and

x² = - 5 ( take the square root of both sides )

x = ±
`√(-5)` = ±
`√(5(-1))` = ±
`√(5)` ×
`√(-1)` = ± i
`√(5)`

Solutions are x = ± i and x = ± i
`√(5)`