((4x-6)/(3x-5))=5x+2

Solution for ((4x-6)/(3x-5))=5x+2 equation:

((4x-6)/(3x-5))=5x+2

We move all terms to the left:

((4x-6)/(3x-5))-(5x+2)=0


Domain of the equation: (3x-5))!=0

x∈R


We get rid of parentheses

((4x-6)/(3x-5))-5x-2=0

We multiply all the terms by the denominator


((4x-6)-5x*(3x-5))-2*(3x-5))=0


We calculate terms in parentheses: +((4x-6)-5x*(3x-5)), so:

(4x-6)-5x*(3x-5)

We multiply parentheses

-15x^2+(4x-6)+25x

We get rid of parentheses

-15x^2+4x+25x-6

We add all the numbers together, and all the variables

-15x^2+29x-6

Back to the equation:

+(-15x^2+29x-6)


We multiply parentheses

(-15x^2+29x-6)-6x-=0

We get rid of parentheses

-15x^2+29x-6x-6-=0

We add all the numbers together, and all the variables

-15x^2+23x=0

a = -15; b = 23; c = 0;
Δ = b2-4ac
Δ = 232-4·(-15)·0
Δ = 529
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{529}=23$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(23)-23}{2*-15}=\frac{-46}{-30}
=1+8/15
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(23)+23}{2*-15}=\frac{0}{-30}
=0
$