3x-7=1/2x+13

Solution for 3x-7=1/2x+13 equation:

3x-7=1/2x+13

We move all terms to the left:

3x-7-(1/2x+13)=0


Domain of the equation: 2x+13)!=0

x∈R


We get rid of parentheses

3x-1/2x-13-7=0

We multiply all the terms by the denominator


3x*2x-13*2x-7*2x-1=0

Wy multiply elements

6x^2-26x-14x-1=0

We add all the numbers together, and all the variables

6x^2-40x-1=0

a = 6; b = -40; c = -1;
Δ = b2-4ac
Δ = -402-4·6·(-1)
Δ = 1624
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:

$\sqrt{\Delta}=\sqrt{1624}=\sqrt{4*406}=\sqrt{4}*\sqrt{406}=2\sqrt{406}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-40)-2\sqrt{406}}{2*6}=\frac{40-2\sqrt{406}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-40)+2\sqrt{406}}{2*6}=\frac{40+2\sqrt{406}}{12}
$