SOLUTION: State both real values of x that satisfy the equation [(3x+4)/(5x+1)]^2+(3x+4)/(5x+1)=12

Question 1096721: State both real values of x that satisfy the equation [(3x+4)/(5x+1)]^2+(3x+4)/(5x+1)=12
Answer by ikleyn(52752) (Show Source): You can put this solution on YOUR website!
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Find all real values of x that satisfy the equation [(3x+4)/(5x+1)]^2+(3x+4)/(5x+1)=12
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 +  = 12.


Introduce new variable u = .  Then your equation becomes this quadratic equation

 = 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   = -4  ====>  3x+4 = (-4)*(5x+1)  ====>  3x+4 = -20x-4  ====>  23x = -8  ====>  x = .


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


Answer.  The original equation has two solutions  x=   and  x= .

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

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