1/5x-23=x+13

Solution for 1/5x-23=x+13 equation:

1/5x-23=x+13

We move all terms to the left:

1/5x-23-(x+13)=0


Domain of the equation: 5x!=0

x!=0/5

x!=0

x∈R


We get rid of parentheses

1/5x-x-13-23=0

We multiply all the terms by the denominator


-x*5x-13*5x-23*5x+1=0

Wy multiply elements

-5x^2-65x-115x+1=0

We add all the numbers together, and all the variables

-5x^2-180x+1=0

a = -5; b = -180; c = +1;
Δ = b2-4ac
Δ = -1802-4·(-5)·1
Δ = 32420
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:

$\sqrt{\Delta}=\sqrt{32420}=\sqrt{4*8105}=\sqrt{4}*\sqrt{8105}=2\sqrt{8105}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-180)-2\sqrt{8105}}{2*-5}=\frac{180-2\sqrt{8105}}{-10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-180)+2\sqrt{8105}}{2*-5}=\frac{180+2\sqrt{8105}}{-10}
$