Question 661525

solve (12x/x-4) - (3x^2/x+4)= (384/x^2-16)


{{{12x/(x - 4) - 3x^2/(x + 4) = 384/(x^2 - 16)}}}


{{{12x/(x - 4) - 3x^2/(x + 4) = 384/(x - 4)(x + 4)}}}


{{{12x(x + 4) - 3x^2(x - 4) = 384}}} ----- Multiplying by LCD, (x - 4)(x + 4)


{{{12x^2 + 48x - 3x^3 + 12x^2 = 384}}}


{{{- 3x^3 + 24x^2 + 48x - 384 = 0}}}


{{{- 3(x^3 - 8x^2 - 16x + 128) = - 3(0)}}} ------ Factoring out GCF, - 3


{{{x^3 - 8x^2 - 16x + 128 = 0}}}


Trying factors of + 128, we can see that one of the roots (solutions) of this equation is 4. Therefore, x - 4 is a factor of this equation.


Dividing the equation, {{{x^3 - 8x^2 - 16x + 128 = 0}}} by (x - 4), we get: {{{x^2 - 4x - 32}}}, and this factors to: (x - 8)(x + 4). 


Therefore, {{{x^3 - 8x^2 - 16x + 128 = 0}}} becomes: {{{(x - 4)(x - 8)(x + 4) = 0}}}. This means that:


x - 4 = 0
{{{highlight(x = 4)}}}


x - 8 = 0
{{{highlight(x = 8)}}}


x + 4 = 0
{{{highlight(x = - 4)}}}


However, {{{x <> 4}}}, or {{{x <> - 4}}}, as either will make the equation UNDEFINED. Therefore, only solution is: {{{highlight_green(x = 8)}}}


After doing all this work, I'm sure you can do the check!!


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