Question 1096721
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{{{highlight(cross(State))}}} Find {{{highlight(cross(both))}}} all real values of x that satisfy the equation [(3x+4)/(5x+1)]^2+(3x+4)/(5x+1)=12
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<pre>
{{{((3x+4)/(5x+1))^2}}} + {{{(3x+4)/(5x+1)}}} = 12.


Introduce new variable u = {{{(3x+4)/(5x+1)}}}.  Then your equation becomes this quadratic equation

{{{u^2 + u - 12}}} = 0.


Factor left side polynomial. You will get

(u+4)*(u-3) = 0,

which has two roots  u= -4  and  u= 3.


a)  Case u= -4:  then  {{{(3x+4)/(5x+1)}}} = -4  ====>  3x+4 = (-4)*(5x+1)  ====>  3x+4 = -20x-4  ====>  23x = -8  ====>  x = {{{-8/23}}}.


b)  Case u= 3:   then  {{{(3x+4)/(5x+1)}}} = 3  ====>  3x+4 = 3*(5x+1)  ====>  3x+4 = 15x+3  ====>  12x = 1  ====>  x = {{{1/12}}}.


<U>Answer</U>.  The original equation has two solutions  x= {{{-8/23}}}  and  x= {{{1/12}}}.
</pre>

Introducing new variable is a standard method of solution non-linear equations like this one.