SOLUTION: (x-5)squared-4(x-5)-21=0

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Question 347066: (x-5)squared-4(x-5)-21=0
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-5%29%5E2-4%28x-5%29-21=0
There's a clever way and a long way to solve this equation. I'll do both.

Clever way, using a temporary variable. This involves noticing that this equation is quadratic in (x-5). It might help you to see this if I use a temporary variable. Let q = x-5. Substituting q for x-5, your equation becomes:
q%5E2+-4q+-21+=+0
This equation is easily solved for q. Just factor it:
%28q-7%29%28q%2B3%29+=+0
Use the Zero Product property:
q-7 = 0 or q+3 = 0
Solving these we get:
q = 7 or q = -3
And last we replace q with x-5:
x-5 = 7 or x-5 = -3
Solving these we get:
x = 12 or x = 2

Clever way without a temporary variable. Eventually you will be able to solve this kind of problem without a temporary variable:
%28x-5%29%5E2-4%28x-5%29-21=0
%28%28x-5%29-7%29%28%28x-5%29%2B3%29+=+0
%28x-12%29%28x-2%29+=+0
x-12 = 0 or x-2 = 0
x = 12 or x = 2
Without the temporary variable this is a very short solution.

The long way to do this problem is to start by simplifying the equation. We'll square (x-5) first:
%28x-5%29%5E2-4%28x-5%29-21=0
x%5E2+-10x+%2B25-4%28x-5%29-21=0
Multiply next:
x%5E2+-10x+%2B25-4x+%2B20-21=0
Combine like terms:
x%5E2+-14x+%2B24=0
Factor:
%28x-12%29%28x-2%29+=+0
Zero Product property:
x-12 = 0 or x-2 = 0
Solve:
x = 12 or x = 2