Question 537580
Although you wrote (according to the rules of algebra) that you were given to solve:
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{{{(1/x^2-3x+2)=(1/x+2)+(5/x^2-4)}}}
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I'm guessing that you really were to solve the following:
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{{{(1/(x^2-3x+2))=(1/(x+2))+(5/(x^2-4))}}}
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If I'm wrong I apologize, and please ignore the following and re-submit your problem.
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Start with:
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{{{(1/(x^2-3x+2))=(1/(x+2))+(5/(x^2-4))}}}
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Where it is possible, factor the denominators. This will give you:
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{{{(1/((x-2)*(x-1)))=(1/(x+2))+(5/((x-2)*(x+2)))}}}
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Next, multiply both sides (all terms) by the three factors common to the denominator. Just multiply each of the terms on both sides by:
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 {{{(x-1)(x+2)(x-2)}}}
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When you do that the equation becomes:
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{{{(((x-1)(x+2)(x-2)*1)/((x-2)*(x-1)))=(((x-1)(x+2)(x-2)*1)/(x+2))+(((x-1)(x+2)(x-2)*5)/((x-2)*(x+2)))}}}
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Now in each term cancel any factors that are in both the numerator and the denominator:
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{{{((cross((x-1))(x+2)cross((x-2))*1)/(cross((x-2))*cross((x-1))))=(((x-1)cross((x+2))(x-2)*1)/cross((x+2)))+(((x-1)cross((x+2))cross((x-2))*5)/(cross((x-2))*cross((x+2))))}}}
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and you are left with:
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{{{(x+2) = (x-1)(x-2)+ 5(x-1)}}}
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Multiply out the two terms on the right side to get:
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{{{x+2 = x^2-3x +2+5x - 5}}}
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On the right side combine the -3x and +5x to get +2x. Also combine the +2 and the -5 to get -3. This reduces the equation to:
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{{{x+2 = x^2 +2x -3}}}
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Then get everything on one side of the equation by subtracting x + 2 from both sides. The equation then becomes:
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{{{0 = x^2 +x -5}}}
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Transpose it to the standard quadratic form:
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{{{x^2 +x-5 =0}}}
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Solve by using the quadratic formula:
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{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
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Recall that a is the multiplier of the {{{x^2}}}. Therefore, a = 1. b is the multiplier of the x. Therefore b also = 1. And c is the constant. And so c = -5. Substituting these values into the quadratic formula results in:
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{{{x = (-(+1) +- sqrt( 1^2-4*1*(-5) ))/(2*1) }}}
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This becomes:
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{{{x = (-1 +- sqrt( 1^2+20 ))/2 }}}
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So the two answers for x are:
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{{{x = (-1 + sqrt( 21 ))/2 }}}
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and
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{{{x = (-1 - sqrt( 21 ))/2 }}}
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Hope this helps you to clear up the places you had trouble with. 
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