SOLUTION: I AM STUCK WITH THIS ONE AND IT IS MOST OFTEN ASKED SO PLEASE HELP ME !! 2 ( x-1 / x+3 ) -7 ( x+3 / x-1 ) = 5

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: I AM STUCK WITH THIS ONE AND IT IS MOST OFTEN ASKED SO PLEASE HELP ME !! 2 ( x-1 / x+3 ) -7 ( x+3 / x-1 ) = 5      Log On


   

Question 846547: I AM STUCK WITH THIS ONE AND IT IS MOST OFTEN ASKED SO PLEASE HELP ME !! 2 ( x-1 / x+3 ) -7 ( x+3 / x-1 ) = 5
Found 2 solutions by Fombitz, josh_jordan:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Multiply both sides by (x-1)(x+3).
2%28x-1%29%5E2-7%28x%2B3%29%5E2=5%28x-1%29%28x%2B3%29
2%28x%5E2-2x%2B1%29-7%28x%5E2%2B6x%2B9%29=5%28x%5E2%2B2x-3%29
2x%5E2-4x%2B2-7x%5E2-42x-63=5x%5E2%2B10x-15
-5x%5E2-46x-61=5x%5E2%2B10x-15
-10x%5E2-56x-46=0
5x%5E2%2B28x%2B23=0
%285x%2B1%29%285x%2B23%29=0
Two solutions:
x%2B1=0
x=-1
x=-1
and
5x%2B23=0
5x=-23
x=-23%2F5

Answer by josh_jordan(263) About Me  (Show Source):
You can put this solution on YOUR website!
2%28%28x-1%29%2F%28x%2B3%29%29-7%28%28x%2B3%29%2F%28x-1%29%29=5
The first thing we need to do is multiply 2 by (x - 1) and multiply 7 by (x + 3):

%28%282x-2%29%2F%28x%2B3%29%29-%28%287x%2B21%29%2F%28x-1%29%29=5
Next, we need to subtract the remaining fractions on the left side of the equation. We do this by multiplying (2x - 2) by (x - 1) and multiplying (7x + 21) by (x + 3) and setting this result in the numerator and setting (x + 3)(x - 1) in the denominator:

----->



Next, after distributing the minus sign to (7x^2 + 42x + 63), combine all of the terms in the numerator and set that above the denominator:

%282x%5E2-4x%2B2-7x%5E2-42x-63%29%2F%28%28x%2B3%29%28x-1%29%29=5 ----->

%28-5x%5E2-46x-61%29%2F%28%28x%2B3%29%28x-1%29%29=5

Next, since this single fraction is set equal to a number, we will cross-multiply the numerator in the fraction, by 1, since 5 = 5/1, and multiply the denominator in the fraction by 5:

-5x%5E2-46x-61=5%28x%2B3%29%28x-1%29 ----->

-5x%5E2-46x-61=5%28x%5E2%2B2x-3%29 ----->

-5x%5E2-46x-61=5x%5E2%2B10x-15

Now, since this is a quadratic equation, we need to move all of the terms to one side of the equation and set the other side of the equation equal to 0:

0=5x%5E2%2B5x%5E2%2B10x%2B46x-15%2B61

Combining like terms gives us:

0=10x%5E2%2B56x%2B46

Now we have this in quadratic form. Let's make this equation easier to factor by factoring out 2, giving us:

0=2%285x%5E2%2B28x%2B23%29

Now, this is easily factorable:

0=2%285x%2B23%29%28x%2B1%29

We can discard the 2 we factored out, and set each of our factors equal to zero to find our x's:

5x%2B23=0 ----->

5x=-23 ----->

x=-23%2F5

AND

x%2B1=0 ----->

x=-1

We now have solved for x.

x = -1 and -23/5

You need to verify both of these values of x work by plugging each value of x into the original equation. Both, in fact, do work, so

x = -1 and -23/5