x=27/x+9

Solution for x=27/x+9 equation:

x=27/x+9

We move all terms to the left:

x-(27/x+9)=0


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

x∈R


We get rid of parentheses

x-27/x-9=0

We multiply all the terms by the denominator


x*x-9*x-27=0

We add all the numbers together, and all the variables

-9x+x*x-27=0

Wy multiply elements

x^2-9x-27=0

a = 1; b = -9; c = -27;
Δ = b2-4ac
Δ = -92-4·1·(-27)
Δ = 189
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:

$\sqrt{\Delta}=\sqrt{189}=\sqrt{9*21}=\sqrt{9}*\sqrt{21}=3\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-3\sqrt{21}}{2*1}=\frac{9-3\sqrt{21}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+3\sqrt{21}}{2*1}=\frac{9+3\sqrt{21}}{2}
$