Question 1188994
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The negative reciprocal of the sum of all values of x that satisfy the equation, (x-sqrt(x+1))/(x+sqrt(x+1)) = 11/5 is...
A) 7 and 1/9 B) -(9/64) C) 1 and 1/8 D) -(1/8) E) 8/9
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<pre>
I will simply solve the given equation 

    {{{(x-sqrt(x+1))/(x+sqrt(x+1))}}} = {{{11/5}}}


and will find its roots. Then I will calculate the negative reciprocal of the sum of the roots.


To solve equation, I will introduce new variable  u = {{{sqrt(x+1)}}},  so  {{{u^2}}} = x+1,  x = {{{u^2 - 1}}}.


With new variable, equation (1)  takes the form


    {{{(u^2 - 1 - u)/(u^2 -1 + u)}}} = {{{11/5}}},

    5*(u^2 - 1 - u) = 11*(u^2 - 1 + u)

    5u^2 - 5 - 5u = 11u^2 - 11 + 11u

    6u^2 + 16u - 6 = 0

    3u^2 + 8u - 3 = 0

    {{{u[1,2]}}} = {{{(-8 +- sqrt((-8)^2 - 4*3*(-3)))/(2*5)}}} = {{{(-8 +- sqrt(64 + 36))/6}}} = {{{(-8 +- 10)/6}}}.


So,  {{{u[1]}}} = -3;  {{{u[2]}}} = {{{1/3}}}.


Thus, there are two solutions for x, and they are  {{{x[1]}}} = {{{u[1]^2-1}}} = {{{(-3)^2-1}}} = 9 - 1 = 8;

                                                   {{{x[2]}}} = {{{u[2]^2-1}}} = {{{(1/3)^2-1}}} = {{{1/9 - 1}}} = {{{-8/9}}}.


The sum of the roots of the original equation is  {{{x[1]}}} + {{{x[2]}}} = 8 - {{{8/9}}} = 7 {{{1/9}}} = {{{64/9}}},

    and the negative reciprocal of it is  {{{-9/64}}}.    <U>ANSWER</U>
</pre>

Solved.