Question

25x^-4-99x^-2-4=0
How do I solve for this?

Answer 1

Given

`25x^(-4) - 99x^(-2) - 4 = 0`

consider substituting
`y=x^(-2)` to get a proper quadratic equation,

`25y^2 - 99y - 4 = 0`

Solve for
`y` ; we can factorize to get

`(25y + 1) (y - 4) = 0`

`25y+1 = 0 \text{ or } y - 4 = 0`

`y = -\frac1{25} \text{ or }y = 4`

Solve for
`x` :

`x^(-2) = -\frac1{25} \text{ or }x^(-2) = 4`

The first equation has no real solution, since
`x^(-2) = \frac1{x^2} > 0` for all non-zero
`x`. Proceeding with the second equation, we get

`x^(-2) = 4 \implies x^2 = \frac14 \implies x = \pm√(\frac14) = \boxed{\pm \frac12}`

If we want to find all complex solutions, we take
`i=√(-1)` so that the first equation above would have led us to

`x^(-2) = -\frac1{25} \implies x^2 = -25 \implies x = \pm√(-25) = \pm5i`