SOLUTION: (x+1)/(x+3) = (x^2-11x)/(x^2+x-6) + (x-3)/(x-2)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: (x+1)/(x+3) = (x^2-11x)/(x^2+x-6) + (x-3)/(x-2)      Log On


   

Question 264983: (x+1)/(x+3) = (x^2-11x)/(x^2+x-6) + (x-3)/(x-2)
Answer by JBarnum(2146) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2B1%29%2F%28x%2B3%29=%28%28x%5E2-11x%29%2F%28x%5E2%2Bx-6%29%29%2B%28x-3%29%2F%28x-2%29
must make the common denominator the same to add/subtract together
just take your time and focus on just the denominators
(x^2+x-6) , (x-2) , (x+3)
.
its common for books to provide the simpilar common denominator in the question.
if you can factor %28x%5E2%2Bx-6%29 it equals %28%28x-2%29%28x%2B3%29%29
if you arent great at factoring then first just try%28%28x-2%29%28x%2B3%29%29 to see if it = %28x%5E2%2Bx-6%29 since it is, you will multiply by the common denominator of%28%28x-2%29%28x%2B3%29%29
.
Now heres what it should look like:
+%28%28x-3%29%2F%28x-2%29%29%28%28%28x-2%29%28x%2B3%29%29%2F1%29%29
Now your ready to do cross cancelations and remember%28x%5E2%2Bx-6%29=%28%28x-2%29%28x%2B3%29%29
%28%28x%2B1%29%28x-2%29%29=%28x%5E2-11x%29%2B%28%28x-3%29%28x%2B3%29%29
Use FOIL method (First Outer Inner Last)
%28x%5E2%2Bx-2x-2%29=%28x%5E2-11x%29%2B%28x%5E2-3x%2B3x-9%29
x%5E2-x-2=x%5E2-11x%2Bx%5E2-9
%28x%5E2-x-2%29=%282x%5E2-11x-9%29
%28x%5E2-x-2%29-%28x%5E2-x-2%29=%282x%5E2-11x-9%29-%28x%5E2-x-2%29
0=%282x%5E2-11x-9%29-%28x%5E2-x-2%29
0=2x%5E2-11x-9-x%5E2%2Bx%2B2%29
0=x%5E2-10x-7%29 Use the Quadratic formula: Ax^2+Bx+C=0 where A=1 B=-10 C=-7
the answer is: 10.6568542494924, -0.656854249492381
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B-7+=+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=%28-10%29%5E2-4%2A1%2A-7=128.

Discriminant d=128 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+128+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+128+%29%29%2F2%5C1+=+10.6568542494924
x%5B2%5D+=+%28-%28-10%29-sqrt%28+128+%29%29%2F2%5C1+=+-0.656854249492381

Quadratic expression 1x%5E2%2B-10x%2B-7 can be factored:
1x%5E2%2B-10x%2B-7+=+1%28x-10.6568542494924%29%2A%28x--0.656854249492381%29
Again, the answer is: 10.6568542494924, -0.656854249492381. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B-7+%29