Question 535600
As you described it, the problem you are trying to solve is:
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{{{x + 5/x = -6}}}
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Get rid of the denominator by multiplying both sides of the equation (all terms) by x and you have:
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{{{(x*x) + x*(5/x) = -6x}}}
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Note that the first term on the left side becomes {{{x^2}}}.
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Also note that in the second term on the left side, you can cancel the multiplier x with the denominator x. After these two actions the equation becomes:
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{{{x^2 + (cross(x)*5)/cross(x) = -6x}}}
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which simplifies to:
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{{{x^2 + 5 = -6x}}}
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Get rid of the -6x on the right side by adding +6x to both sides to get:
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{{{x^2 + 6x + 5 = 0}}}
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This quadratic equation can be factored as shown below:
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{{{(x + 5)*(x + 1) = 0}}}
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Note that this equation will be true if either of the factors is equal to zero. If either factor is equal to zero the left side will have a multiplication by zero and will, therefore, equal the zero on the right side. 
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So we can solve this equation by finding what numbers will make the two factors equal to zero. Do this by setting each factor (one at a time) equal to zero to get:
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{{{x + 5 = 0}}} which is solved by subtracting 5 from both sides to get x = -5
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and
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{{{x + 1 = 0}}} which is solved by subtracting 1 from both sides to get x = -1
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Your problem has two correct answers. They are x = -5 and x = -1.
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You can check these answers by returning to the original problem and substituting these values one at a time for x. 
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{{{x + 5/x = -6}}}
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set x = -5 and you have:
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{{{-5 + 5/-5 = -6}}} and this simplifies to {{{-5 -1 = -6}}} which is true.
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Then set x = -1 and you have:
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{{{-1 + 5/-1 = -6}}} and this simplifies to {{{-1 -5 = -6}}} which is also true.
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Both answers are OK because they make the original problem true.
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Hope this helps you to understand the problem.
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