3x=7/6x+1

Solution for 3x=7/6x+1 equation:

3x=7/6x+1

We move all terms to the left:

3x-(7/6x+1)=0


Domain of the equation: 6x+1)!=0

x∈R


We get rid of parentheses

3x-7/6x-1=0

We multiply all the terms by the denominator


3x*6x-1*6x-7=0

Wy multiply elements

18x^2-6x-7=0

a = 18; b = -6; c = -7;
Δ = b2-4ac
Δ = -62-4·18·(-7)
Δ = 540
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:

$\sqrt{\Delta}=\sqrt{540}=\sqrt{36*15}=\sqrt{36}*\sqrt{15}=6\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-6\sqrt{15}}{2*18}=\frac{6-6\sqrt{15}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+6\sqrt{15}}{2*18}=\frac{6+6\sqrt{15}}{36}
$