```Question 325467

{{{3x/(x-4) + x/(x+4) = 24/((x-4)(x+4))}}} Factor {{{x^2-16}}} to get {{{(x-4)(x+4)}}}

{{{3x(x+4) + x(x-4) = 24}}} Multiply every term by the LCD {{{(x-4)(x+4)}}} to clear out the fractions.

{{{3x^2+12x+ x^2-4x = 24}}} Distribute.

{{{3x^2+12x+x^2-4x-24=0}}} Subtract 24 from both sides.

{{{4x^2+8x-24=0}}} Combine like terms.

Notice that the quadratic {{{4x^2+8x-24}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=4}}}, {{{B=8}}}, and {{{C=-24}}}

Let's use the quadratic formula to solve for "x":

{{{x = (-(8) +- sqrt( (8)^2-4(4)(-24) ))/(2(4))}}} Plug in  {{{A=4}}}, {{{B=8}}}, and {{{C=-24}}}

{{{x = (-8 +- sqrt( 64-4(4)(-24) ))/(2(4))}}} Square {{{8}}} to get {{{64}}}.

{{{x = (-8 +- sqrt( 64--384 ))/(2(4))}}} Multiply {{{4(4)(-24)}}} to get {{{-384}}}

{{{x = (-8 +- sqrt( 64+384 ))/(2(4))}}} Rewrite {{{sqrt(64--384)}}} as {{{sqrt(64+384)}}}

{{{x = (-8 +- sqrt( 448 ))/(2(4))}}} Add {{{64}}} to {{{384}}} to get {{{448}}}

{{{x = (-8 +- sqrt( 448 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}.

{{{x = (-8 +- 8*sqrt(7))/(8)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)

{{{x = (-8)/(8) +- (8*sqrt(7))/(8)}}} Break up the fraction.

{{{x = -1 +- sqrt(7)}}} Reduce.

{{{x = -1+sqrt(7)}}} or {{{x = -1-sqrt(7)}}} Break up the expression.

So the solutions are {{{x = -1+sqrt(7)}}} or {{{x = -1-sqrt(7)}}} ```