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

#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#

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#