3+4z=z^2=0

Solution for 3+4z=z^2=0 equation:

3+4z=z^2=0

We move all terms to the left:

3+4z-(z^2)=0

determiningTheFunctionDomain
-z^2+4z+3=0

We add all the numbers together, and all the variables

-1z^2+4z+3=0

a = -1; b = 4; c = +3;
Δ = b2-4ac
Δ = 42-4·(-1)·3
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{7}}{2*-1}=\frac{-4-2\sqrt{7}}{-2}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{7}}{2*-1}=\frac{-4+2\sqrt{7}}{-2}
$