.5z^2-z=13

Solution for .5z^2-z=13 equation:

.5z^2-z=13

We move all terms to the left:

.5z^2-z-(13)=0

We add all the numbers together, and all the variables

.5z^2-1z-13=0

a = .5; b = -1; c = -13;
Δ = b2-4ac
Δ = -12-4·.5·(-13)
Δ = 27
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:

$\sqrt{\Delta}=\sqrt{27}=\sqrt{9*3}=\sqrt{9}*\sqrt{3}=3\sqrt{3}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3\sqrt{3}}{2*.5}=\frac{1-3\sqrt{3}}{1}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3\sqrt{3}}{2*.5}=\frac{1+3\sqrt{3}}{1}
$