z/2=4z^2-1

Solution for z/2=4z^2-1 equation:

z/2=4z^2-1

We move all terms to the left:

z/2-(4z^2-1)=0

We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

-4z^2*2+z+2=0

Wy multiply elements

-8z^2+z+2=0

a = -8; b = 1; c = +2;
Δ = b2-4ac
Δ = 12-4·(-8)·2
Δ = 65
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{65}}{2*-8}=\frac{-1-\sqrt{65}}{-16}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{65}}{2*-8}=\frac{-1+\sqrt{65}}{-16}
$