1/2z=53z=42

Solution for 1/2z=53z=42 equation:

1/2z=53z=42

We move all terms to the left:

1/2z-(53z)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


We add all the numbers together, and all the variables

-53z+1/2z=0

We multiply all the terms by the denominator


-53z*2z+1=0

Wy multiply elements

-106z^2+1=0

a = -106; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-106)·1
Δ = 424
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:

$\sqrt{\Delta}=\sqrt{424}=\sqrt{4*106}=\sqrt{4}*\sqrt{106}=2\sqrt{106}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{106}}{2*-106}=\frac{0-2\sqrt{106}}{-212}
=-\frac{2\sqrt{106}}{-212}
=-\frac{\sqrt{106}}{-106}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{106}}{2*-106}=\frac{0+2\sqrt{106}}{-212}
=\frac{2\sqrt{106}}{-212}
=\frac{\sqrt{106}}{-106}
$