x-4=1/4x+5

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

x-4=1/4x+5

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

4x^2-20x-16x-1=0

We add all the numbers together, and all the variables

4x^2-36x-1=0

a = 4; b = -36; c = -1;
Δ = b2-4ac
Δ = -362-4·4·(-1)
Δ = 1312
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:

$\sqrt{\Delta}=\sqrt{1312}=\sqrt{16*82}=\sqrt{16}*\sqrt{82}=4\sqrt{82}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-4\sqrt{82}}{2*4}=\frac{36-4\sqrt{82}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+4\sqrt{82}}{2*4}=\frac{36+4\sqrt{82}}{8}
$