x=1/6x+9

Solution for x=1/6x+9 equation:

x=1/6x+9

We move all terms to the left:

x-(1/6x+9)=0


Domain of the equation: 6x+9)!=0

x∈R


We get rid of parentheses

x-1/6x-9=0

We multiply all the terms by the denominator


x*6x-9*6x-1=0

Wy multiply elements

6x^2-54x-1=0

a = 6; b = -54; c = -1;
Δ = b2-4ac
Δ = -542-4·6·(-1)
Δ = 2940
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:

$\sqrt{\Delta}=\sqrt{2940}=\sqrt{196*15}=\sqrt{196}*\sqrt{15}=14\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-14\sqrt{15}}{2*6}=\frac{54-14\sqrt{15}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+14\sqrt{15}}{2*6}=\frac{54+14\sqrt{15}}{12}
$