Question

How do you factor `x^-4 -13x^-2 +36 =0`?

Answer 1

`x=+-1/2, +-1/3`

Multiply by `x^4` to give `36x^4-13x^2+1`
Solve using the quadratic formula for `x^2` `x^2=(13+-sqrt(13^2-4.36))/(2.36)=(13+-sqrt(169-144))/72=(13+-sqrt(25))/72=18/72,8/72=1/4,1/9`
Taking square roots gives `x=+-1/2, +-1/3`

Answer 2

Terminology:
You can factor the expression on the left. You can solve the equation. You can solve an equation by factoring.
(But you don't really factor an equation.)

To factor: `x^(-4)-13x^(-2)+36`

Notice that `x^(-4) = (x^(-2))^2` (The variable expression in the first term is the square of the one in the second term.)

So, is we use a new variable we'll have a quadratic expression.

Let `u=x^(-2)`. This makes `u^2 = x^(-4)`, so the expression becomes:

`u^2-13u+36` which can be factored:

`(u-4)(u-9)` Now go back to `x`'s

`x^(-4)-13x^(-2)+36 = (x^(-2)-4)(x^(-2)-9)`

To solve by factoring: `x^(-4)-13x^(-2)+36=0`

Factor as above, so the question becomes:

Solve: `(x^(-2)-4)(x^(-2)-9) = 0`

So we need: `(x^(-2)-4)=0` or `(x^(-2)-9) = 0`

`x^(-2)-4=0` `color(white)"sssss"` or `color(white)"sssss"` `x^(-2)-9 = 0`

`x^(-2) = 4` `color(white)"ssssssssss"` or `color(white)"sssss"` `x^(-2) = 9`

`1/x^2 = 4` `color(white)"ssssssssss"` or `color(white)"sssss"` `1/x^2 = 9`

`1/4 = x^2` `color(white)"sssssssssss"` or `color(white)"sssss"` `1/9 = x^2`

`x = +- 1/2` `color(white)"ssssssss"` or `color(white)"sssss"` `x= +- 1/3`

There are four solutions: `-1/2, 1/2, -1/3, 1/3`