SOLUTION: Please help me solve these rational equation: x^4+2x^2= x/x+2+3/x+3=13/x^2+5x+6

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Question 117610: Please help me solve these rational equation:
x^4+2x^2=
x/x+2+3/x+3=13/x^2+5x+6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#1

Do you want to factor x^4+2x^2?

x%5E4%2B2x%5E2 Start with the given expression


x%5E2%28x%5E2%2B2%29 Factor out the GCF x%5E2




#2

x%2F%28x%2B2%29%2B3%2F%28x%2B3%29=13%2F%28x%5E2%2B5x%2B6%29 Start with the given expression


x%2F%28x%2B2%29%2B3%2F%28x%2B3%29=13%2F%28%28x%2B2%29%28x%2B3%29%29 Factor the denominator on the right side


Multiply both sides by the LCD %28x%2B2%29%28x%2B3%29. Notice how every denominator is eliminated


x%28x%2B3%29%2B3%28x%2B2%29=13 Distribute and multiply


x%5E2%2B3x%2B3x%2B6=13 Distribute again


x%5E2%2B3x%2B3x%2B6-13=0 Subtract 13 from both sides



x%5E2%2B6x-7=0 Combine like terms


%28x%2B7%29%28x-1%29=0 Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
x%2B7=0 or x-1=0

x=-7 or x=1 Now solve for x in each case


So our answer is
x=-7 or x=1



Notice if we graph y=x%2F%28x%2B2%29%2B3%2F%28x%2B3%29 and y=13%2F%28x%5E2%2B5x%2B6%29 we can see that they intersect at x=-7 and x=1. So this visually verifies our answer. note: Ignore the vertical lines. They are not part of the graph.