Question 94432
{{{9/(x-5) -1=8/(x+5)}}}



{{{(x-5)(x+5)(9/(x-5) -1)=(x-5)(x+5)(8/(x+5))}}} Multiply by the LCD {{{(x-5)(x+5)}}}


{{{9(x+5) -(x-5)(x+5)=8(x-5)}}} Distribute and simplify


{{{9x+45 -(x-5)(x+5)=8x-40}}} Distribute again


{{{9x+45 -(x^2-25)=8x-40}}} Foil


{{{9x+45 -x^2+25=8x-40}}} Distribute the negative


{{{9x+45 -x^2+25-8x+40=0}}} Get all terms to one side


{{{-x^2+x+110=0}}} Combine like terms




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



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{-x^2+x+110=0}}} ( notice {{{a=-1}}}, {{{b=1}}}, and {{{c=110}}})


{{{x = (-1 +- sqrt( (1)^2-4*-1*110 ))/(2*-1)}}} Plug in a=-1, b=1, and c=110




{{{x = (-1 +- sqrt( 1-4*-1*110 ))/(2*-1)}}} Square 1 to get 1  




{{{x = (-1 +- sqrt( 1+440 ))/(2*-1)}}} Multiply {{{-4*110*-1}}} to get {{{440}}}




{{{x = (-1 +- sqrt( 441 ))/(2*-1)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-1 +- 21)/(2*-1)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (-1 +- 21)/-2}}} Multiply 2 and -1 to get -2


So now the expression breaks down into two parts


{{{x = (-1 + 21)/-2}}} or {{{x = (-1 - 21)/-2}}}


Lets look at the first part:


{{{x=(-1 + 21)/-2}}}


{{{x=20/-2}}} Add the terms in the numerator

{{{x=-10}}} Divide


So one answer is

{{{x=-10}}}




Now lets look at the second part:


{{{x=(-1 - 21)/-2}}}


{{{x=-22/-2}}} Subtract the terms in the numerator

{{{x=11}}} Divide


So another answer is

{{{x=11}}}


So our solutions are:

{{{x=-10}}} or {{{x=11}}}


Notice when we graph {{{-x^2+x+110}}}, we get:


{{{ graph( 500, 500, -20, 21, -20, 21,-1*x^2+1*x+110) }}}


and we can see that the roots are {{{x=-10}}} and {{{x=11}}}. This verifies our answer




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Check:

Let's check the solution x=-10


{{{9/(x-5) -1=8/(x+5)}}} Start with the given equation


{{{9/(-10-5) -1=8/(-10+5)}}} Plug in x=-10


{{{9/(-15) -1=8/(-5)}}} Simplify the denominators


{{{-3/5 -1=-8/5}}} Reduce



{{{-3/5 -5/5=-8/5}}} Rewrite -1 as {{{-5/5}}}


{{{-8/5=-8/5}}} Combine the fractions. Since the two sides of the equation are equal, this verifies the solution x=-10








Let's check the solution x=11


{{{9/(x-5) -1=8/(x+5)}}} Start with the given equation


{{{9/(11-5) -1=8/(11+5)}}} Plug in x=11


{{{9/(6) -1=8/(16)}}} Simplify the denominators


{{{3/2 -1=1/2}}} Reduce



{{{3/2 -2/2=1/2}}} Rewrite -1 as {{{-2/2}}}


{{{1/2=1/2}}} Combine the fractions. Since the two sides of the equation are equal, this verifies the solution x=11