((5x+7)/x-1)=3x+2

Solution for ((5x+7)/x-1)=3x+2 equation:

((5x+7)/x-1)=3x+2

We move all terms to the left:

((5x+7)/x-1)-(3x+2)=0


Domain of the equation: x-1)!=0

x∈R


We get rid of parentheses

((5x+7)/x-1)-3x-2=0

We multiply all the terms by the denominator


((5x+7)-3x*x-1)-2*x-1)=0


We calculate terms in parentheses: +((5x+7)-3x*x-1), so:

(5x+7)-3x*x-1

Wy multiply elements

-3x^2+(5x+7)-1

We get rid of parentheses

-3x^2+5x+7-1

We add all the numbers together, and all the variables

-3x^2+5x+6

Back to the equation:

+(-3x^2+5x+6)


We add all the numbers together, and all the variables

(-3x^2+5x+6)-2x=0

We get rid of parentheses

-3x^2+5x-2x+6=0

We add all the numbers together, and all the variables

-3x^2+3x+6=0

a = -3; b = 3; c = +6;
Δ = b2-4ac
Δ = 32-4·(-3)·6
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-9}{2*-3}=\frac{-12}{-6}
=+2
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+9}{2*-3}=\frac{6}{-6}
=-1
$