Question

What are the real and complex solutions to the following equation? -2x^4+2x^2+4=0

Show all steps.

u=±1i or u=±√2

u=±2i or u=±√2

Answer 1

x = ± i, x = ±
`√(2)`

Explanation:

Given

- 2
`x^(4)` + 2x² + 4 = 0 ( divide through by - 2 )

`x^(4)` - x² - 2 = 0

Use the substitution x² = u , then

u² - u - 2 = 0 ← in standard form

(u - 2)(u + 1) = 0 ← in factored form

Equate each factor to zero and solve for u

u - 2 = 0 ⇒ u = 2

u + 1 = 0 ⇒ u = - 1

Change back into terms of x, that is

x² = 2 ( take the square root of both sides )

x = ±
`√(2)`

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

x = ±
`√(-1)` = ± i

x = ±
`√(2)` ← real solutions

x = ± i ← complex solutions