(z+40)(z-20)=90

Solution for (z+40)(z-20)=90 equation:

(z+40)(z-20)=90

We move all terms to the left:

(z+40)(z-20)-(90)=0

We multiply parentheses ..

(+z^2-20z+40z-800)-90=0

We get rid of parentheses

z^2-20z+40z-800-90=0

We add all the numbers together, and all the variables

z^2+20z-890=0

a = 1; b = 20; c = -890;
Δ = b2-4ac
Δ = 202-4·1·(-890)
Δ = 3960
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:

$\sqrt{\Delta}=\sqrt{3960}=\sqrt{36*110}=\sqrt{36}*\sqrt{110}=6\sqrt{110}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-6\sqrt{110}}{2*1}=\frac{-20-6\sqrt{110}}{2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+6\sqrt{110}}{2*1}=\frac{-20+6\sqrt{110}}{2}
$