z.9=27/z

Solution for z.9=27/z equation:

z.9=27/z

We move all terms to the left:

z.9-(27/z)=0


Domain of the equation: z)!=0

z!=0/1

z!=0

z∈R


We add all the numbers together, and all the variables

z.9-(+27/z)=0

We get rid of parentheses

z.9-27/z=0

We multiply all the terms by the denominator


(z.9)*z-27=0

We add all the numbers together, and all the variables

(+z.9)*z-27=0

We multiply parentheses

z^2-27=0

a = 1; b = 0; c = -27;
Δ = b2-4ac
Δ = 02-4·1·(-27)
Δ = 108
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-6\sqrt{3}}{2*1}=\frac{0-6\sqrt{3}}{2}
=-\frac{6\sqrt{3}}{2}
=-3\sqrt{3}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+6\sqrt{3}}{2*1}=\frac{0+6\sqrt{3}}{2}
=\frac{6\sqrt{3}}{2}
=3\sqrt{3}
$