x=1/4x+6

Solution for x=1/4x+6 equation:

x=1/4x+6

We move all terms to the left:

x-(1/4x+6)=0


Domain of the equation: 4x+6)!=0

x∈R


We get rid of parentheses

x-1/4x-6=0

We multiply all the terms by the denominator


x*4x-6*4x-1=0

Wy multiply elements

4x^2-24x-1=0

a = 4; b = -24; c = -1;
Δ = b2-4ac
Δ = -242-4·4·(-1)
Δ = 592
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:

$\sqrt{\Delta}=\sqrt{592}=\sqrt{16*37}=\sqrt{16}*\sqrt{37}=4\sqrt{37}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-4\sqrt{37}}{2*4}=\frac{24-4\sqrt{37}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+4\sqrt{37}}{2*4}=\frac{24+4\sqrt{37}}{8}
$