SOLUTION: Factor this expression as a difference of squares. a^2+4a-b^2

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Question 1047131: Factor this expression as a difference of squares. a^2+4a-b^2




Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Already answered this question.

John

My calculator said it, I believe it, that settles it

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
This cannot be factored using only rational expressions, 
but if you allow square roots in the factorization, then 
there are more than one way to do this, and each way gives
a different expression. 

Here's the first way:

a%5E2%2B4a-b%5E2

Write aČ+4a as its square root squared:

%28sqrt%28a%5E2%2B4a%29%29%5E2-b%5E2

Then factor it as the difference of two squares:

%28sqrt%28a%5E2%2B4a%29-b%29%28sqrt%28a%5E2%2B4a%29%2Bb%29

Here's a second way:

a%5E2%2B4a-b%5E2

Swap the second and third terms:

a%5E2-b%5E2%2B4a

Factor out a - from the 2nd and 3rd terms:

a%5E2-%28b%5E2-4a%29

Write the expression in parentheses as the 
square of its square root:

a%5E2-%28sqrt%28b%5E2-4a%29%29%5E2

Then factor as the difference of two squares:

%28a-sqrt%28b%5E2-4a%29%29%28a%2Bsqrt%28b%5E2-4a%29%29


Here's a third way:

a%5E2%2B4a-b%5E2

Add +4 after the 4a and subtract 4 after the -b^2

a%5E2%2B4a%2B4-b%5E2-4

Factor the first three terms as a perfect square trinomial.
Factor a negative out of the last two terms:

%28a%2B2%29%5E2-%28b%5E2%2B4%29

Then write the expression in parentheses after the - sign 
as the square of its square root.

%28a%2B2%29%5E2-%28sqrt%28b%5E2%2B4%29%29%5E2

Then as

%28%28a%2B2%29-sqrt%28b%5E2%2B4%29%29%28%28a%2B2%29%2Bsqrt%28b%5E2%2B4%29%29

There are other ways as well, but they'll always involve
square roots.  

Edwin