x-13=1/4x+5

Solution for x-13=1/4x+5 equation:

x-13=1/4x+5

We move all terms to the left:

x-13-(1/4x+5)=0


Domain of the equation: 4x+5)!=0

x∈R


We get rid of parentheses

x-1/4x-5-13=0

We multiply all the terms by the denominator


x*4x-5*4x-13*4x-1=0

Wy multiply elements

4x^2-20x-52x-1=0

We add all the numbers together, and all the variables

4x^2-72x-1=0

a = 4; b = -72; c = -1;
Δ = b2-4ac
Δ = -722-4·4·(-1)
Δ = 5200
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:

$\sqrt{\Delta}=\sqrt{5200}=\sqrt{400*13}=\sqrt{400}*\sqrt{13}=20\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-72)-20\sqrt{13}}{2*4}=\frac{72-20\sqrt{13}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-72)+20\sqrt{13}}{2*4}=\frac{72+20\sqrt{13}}{8}
$