12+1/6x=17-23x

Solution for 12+1/6x=17-23x equation:

12+1/6x=17-23x

We move all terms to the left:

12+1/6x-(17-23x)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R


We add all the numbers together, and all the variables

1/6x-(-23x+17)+12=0

We get rid of parentheses

1/6x+23x-17+12=0

We multiply all the terms by the denominator


23x*6x-17*6x+12*6x+1=0

Wy multiply elements

138x^2-102x+72x+1=0

We add all the numbers together, and all the variables

138x^2-30x+1=0

a = 138; b = -30; c = +1;
Δ = b2-4ac
Δ = -302-4·138·1
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{87}}{2*138}=\frac{30-2\sqrt{87}}{276}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{87}}{2*138}=\frac{30+2\sqrt{87}}{276}
$