8=4(z-78)z=

Solution for 8=4(z-78)z= equation:

8=4(z-78)z=

We move all terms to the left:

8-(4(z-78)z)=0


We calculate terms in parentheses: -(4(z-78)z), so:

4(z-78)z

We multiply parentheses

4z^2-312z

Back to the equation:

-(4z^2-312z)


We get rid of parentheses

-4z^2+312z+8=0

a = -4; b = 312; c = +8;
Δ = b2-4ac
Δ = 3122-4·(-4)·8
Δ = 97472
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:

$\sqrt{\Delta}=\sqrt{97472}=\sqrt{64*1523}=\sqrt{64}*\sqrt{1523}=8\sqrt{1523}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(312)-8\sqrt{1523}}{2*-4}=\frac{-312-8\sqrt{1523}}{-8}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(312)+8\sqrt{1523}}{2*-4}=\frac{-312+8\sqrt{1523}}{-8}
$