3x-7-2/(9x-6)=15

Solution for 3x-7-2/(9x-6)=15 equation:

3x-7-2/(9x-6)=15

We move all terms to the left:

3x-7-2/(9x-6)-(15)=0


Domain of the equation: (9x-6)!=0

We move all terms containing x to the left, all other terms to the right

9x!=6

x!=6/9

x!=2/3

x∈R


We add all the numbers together, and all the variables

3x-2/(9x-6)-22=0

We multiply all the terms by the denominator


3x*(9x-6)-22*(9x-6)-2=0

We multiply parentheses

27x^2-18x-198x+132-2=0

We add all the numbers together, and all the variables

27x^2-216x+130=0

a = 27; b = -216; c = +130;
Δ = b2-4ac
Δ = -2162-4·27·130
Δ = 32616
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{32616}=\sqrt{36*906}=\sqrt{36}*\sqrt{906}=6\sqrt{906}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-216)-6\sqrt{906}}{2*27}=\frac{216-6\sqrt{906}}{54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-216)+6\sqrt{906}}{2*27}=\frac{216+6\sqrt{906}}{54}
$