z^2+8z-8/27=0

Solution for z^2+8z-8/27=0 equation:

z^2+8z-8/27=0

We multiply all the terms by the denominator


z^2*27+8z*27-8=0

Wy multiply elements

27z^2+216z-8=0

a = 27; b = 216; c = -8;
Δ = b2-4ac
Δ = 2162-4·27·(-8)
Δ = 47520
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:

$\sqrt{\Delta}=\sqrt{47520}=\sqrt{144*330}=\sqrt{144}*\sqrt{330}=12\sqrt{330}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(216)-12\sqrt{330}}{2*27}=\frac{-216-12\sqrt{330}}{54}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(216)+12\sqrt{330}}{2*27}=\frac{-216+12\sqrt{330}}{54}
$