SOLUTION: Solve the equation by using the LCD. I don't know how to do this one, it look confusing (5) / (x^2+x-6) = (2) + (x-3) / (x-2)

Algebra ->  Rational-functions -> SOLUTION: Solve the equation by using the LCD. I don't know how to do this one, it look confusing (5) / (x^2+x-6) = (2) + (x-3) / (x-2)      Log On


   

Question 1031718: Solve the equation by using the LCD.
I don't know how to do this one, it look confusing
(5) / (x^2+x-6) = (2) + (x-3) / (x-2)
Found 2 solutions by Alan3354, MathTherapy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
5/(x^2+x-6) = 2 + (x-3)/(x-2)
x^2+x-6 can be factored.
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5%2F%28%28x%2B3%29%2A%28x-2%29%29+=+2+%2B+%28x-3%29%2F%28x-2%29
LCD = (x+3)*(x-2)
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5%2F%28%28x%2B3%29%2A%28x-2%29%29+=+2+%2B+%28x-3%29%2A%28x%2B3%29%2FLCD
Multiply thru by DEN, the LCD
5 = 2*(x^2+x-6) + (x-3)*(x+3)
5 = 3x^2 + 2x - 21
3x^2 + 2x - 26 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B2x%2B-26+=+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%2A3%2A-26=316.

Discriminant d=316 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+316+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+316+%29%29%2F2%5C3+=+2.6293981391052
x%5B2%5D+=+%28-%282%29-sqrt%28+316+%29%29%2F2%5C3+=+-3.29606480577186

Quadratic expression 3x%5E2%2B2x%2B-26 can be factored:
3x%5E2%2B2x%2B-26+=+%28x-2.6293981391052%29%2A%28x--3.29606480577186%29
Again, the answer is: 2.6293981391052, -3.29606480577186. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B2%2Ax%2B-26+%29

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Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the equation by using the LCD.
I don't know how to do this one, it look confusing
(5) / (x^2+x-6) = (2) + (x-3) / (x-2)
5%2F%28x%5E2+%2B+x+-+6%29+=+2+%2B+%28x+-+3%29%2F%28x+-+2%29

5%2F%28%28x+-+2%29%28x+%2B+3%29%29+=+2+%2B+%28x+-+3%29%2F%28x+-+2%29 ------- Factoring 1st-fraction denominator: x%5E2+%2B+x+-+6

5 = 2(x - 2)(x + 3) + (x - 3)(x + 3) ------ Multiplying by LCD, (x - 2)(x + 3)

5+=+2%28x%5E2+%2B+x+-+6%29+%2B+x%5E2+-+9

5+=+2x%5E2+%2B+2x+-+12+%2B+x%5E2+-+9

5+=+2x%5E2+%2B+x%5E2+%2B+2x+-+12+-+9 

0+=+3x%5E2+%2B+2x+-+26

Using the quadratic equation formula, or completing the square should result in: