210+30z=z2+6z+27

Solution for 210+30z=z2+6z+27 equation:

210+30z=z2+6z+27

We move all terms to the left:

210+30z-(z2+6z+27)=0

We add all the numbers together, and all the variables

-(+z^2+6z+27)+30z+210=0

We get rid of parentheses

-z^2-6z+30z-27+210=0

We add all the numbers together, and all the variables

-1z^2+24z+183=0

a = -1; b = 24; c = +183;
Δ = b2-4ac
Δ = 242-4·(-1)·183
Δ = 1308
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:

$\sqrt{\Delta}=\sqrt{1308}=\sqrt{4*327}=\sqrt{4}*\sqrt{327}=2\sqrt{327}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-2\sqrt{327}}{2*-1}=\frac{-24-2\sqrt{327}}{-2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+2\sqrt{327}}{2*-1}=\frac{-24+2\sqrt{327}}{-2}
$