SOLUTION: simplify/Reduce 16 4 _______ - ___ x^2+2x-3 x+3

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Question 31602: simplify/Reduce
16 4
_______ - ___
x^2+2x-3 x+3
Answer by mbarugel(146) About Me  (Show Source):
You can put this solution on YOUR website!
Hello!
We can start by finding the roots of the quadratic equation in the denominator of the 1st term:
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B2x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A1%2A-3=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+16+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+16+%29%29%2F2%5C1+=+1
x%5B2%5D+=+%28-%282%29-sqrt%28+16+%29%29%2F2%5C1+=+-3

Quadratic expression 1x%5E2%2B2x%2B-3 can be factored:
1x%5E2%2B2x%2B-3+=+%28x-1%29%2A%28x--3%29
Again, the answer is: 1, -3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-3+%29


The roots are 1 and -3. This implies that that quadratic equation can be written as:
x%5E2%2B2x-3+=+%28x-1%29%28x+-%28-3%29%29+=+%28x-1%29%28x%2B3%29
So your expression becomes:
16%2F%28%28x-1%29%28x%2B3%29%29+-+4%2F%28x%2B3%29
But since 4 is a factor of 16 and (x+3) is a factor of (x-1)(x+3), then we can take 4%2F%28x%2B3%29 as a common factor, getting:
%284%2F%28x%2B3%29%29%28+4%2F%28x-1%29+-+1%29
Furthermore, we can add the fractions inside the parenthesis, getting:
%284%2F%28x%2B3%29%29%28%28+4+-+%28x-1%29%29%2F%28x-1%29%29
%284%2F%28x%2B3%29%29%28%28+5-x%29%2F%28x-1%29%29
And that's as far as we can go.

I hope this helps!
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