(z+2)(z+6)=-3

Solution for (z+2)(z+6)=-3 equation:

(z+2)(z+6)=-3

We move all terms to the left:

(z+2)(z+6)-(-3)=0

We add all the numbers together, and all the variables

(z+2)(z+6)+3=0

We multiply parentheses ..

(+z^2+6z+2z+12)+3=0

We get rid of parentheses

z^2+6z+2z+12+3=0

We add all the numbers together, and all the variables

z^2+8z+15=0

a = 1; b = 8; c = +15;
Δ = b2-4ac
Δ = 82-4·1·15
Δ = 4
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{4}=2$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-2}{2*1}=\frac{-10}{2}
=-5
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+2}{2*1}=\frac{-6}{2}
=-3
$