4z2-2z=102

Solution for 4z2-2z=102 equation:

4z^2-2z=102

We move all terms to the left:

4z^2-2z-(102)=0

a = 4; b = -2; c = -102;
Δ = b2-4ac
Δ = -22-4·4·(-102)
Δ = 1636
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:

$\sqrt{\Delta}=\sqrt{1636}=\sqrt{4*409}=\sqrt{4}*\sqrt{409}=2\sqrt{409}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{409}}{2*4}=\frac{2-2\sqrt{409}}{8}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{409}}{2*4}=\frac{2+2\sqrt{409}}{8}
$