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#
