1/2x-13=23+5x

Solution for 1/2x-13=23+5x equation:

1/2x-13=23+5x

We move all terms to the left:

1/2x-13-(23+5x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

1/2x-(5x+23)-13=0

We get rid of parentheses

1/2x-5x-23-13=0

We multiply all the terms by the denominator


-5x*2x-23*2x-13*2x+1=0

Wy multiply elements

-10x^2-46x-26x+1=0

We add all the numbers together, and all the variables

-10x^2-72x+1=0

a = -10; b = -72; c = +1;
Δ = b2-4ac
Δ = -722-4·(-10)·1
Δ = 5224
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:

$\sqrt{\Delta}=\sqrt{5224}=\sqrt{4*1306}=\sqrt{4}*\sqrt{1306}=2\sqrt{1306}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-72)-2\sqrt{1306}}{2*-10}=\frac{72-2\sqrt{1306}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-72)+2\sqrt{1306}}{2*-10}=\frac{72+2\sqrt{1306}}{-20}
$