Question 1127134: Solve for x by factoring. Show all real solutions.
Found 4 solutions by mhusband, josgarithmetic, ikleyn, greenestamps:
Answer by mhusband(3)   (Show Source):
Answer by josgarithmetic(39613) (Show Source):
Answer by ikleyn(52754)  (Show Source):
You can put this solution on YOUR website! .
In this case, factoring works this way
 =  . (1)
The person who has some minimal experience in factoring, will find this decomposition easy.
But if you have no such experience, then you can introduce new variable y =  ,
and then your polynomial becomes
=  .
It is very well familiar quadratic polynomial of "y", and therefore you can factor it quickly
 = (y+8)*(y-1).
Returning then from "y" to , you get this decomposition (1).
In any case, when you get this decomposition
= ,
you easily get the roots:
a) The factor  gives you the root = -8, or x=  = -2, and
b) The factor  gives you the root = 1, or x=  = 1.
Solved.
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Introducing new variables, as I showed you in my post, often can help you in many other complicated cases.
It is very useful trick and a method.
As mathematicians say, introducing new variable/variables allows reducing the equation degree, which always facilitates the solution.
Answer by greenestamps(13195)  (Show Source):
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