1/2x-13=3x+8

Solution for 1/2x-13=3x+8 equation:

1/2x-13=3x+8

We move all terms to the left:

1/2x-13-(3x+8)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x-8-13=0

We multiply all the terms by the denominator


-3x*2x-8*2x-13*2x+1=0

Wy multiply elements

-6x^2-16x-26x+1=0

We add all the numbers together, and all the variables

-6x^2-42x+1=0

a = -6; b = -42; c = +1;
Δ = b2-4ac
Δ = -422-4·(-6)·1
Δ = 1788
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:

$\sqrt{\Delta}=\sqrt{1788}=\sqrt{4*447}=\sqrt{4}*\sqrt{447}=2\sqrt{447}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{447}}{2*-6}=\frac{42-2\sqrt{447}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{447}}{2*-6}=\frac{42+2\sqrt{447}}{-12}
$