5*4z-1/2z=234

Solution for 5*4z-1/2z=234 equation:

5*4z-1/2z=234

We move all terms to the left:

5*4z-1/2z-(234)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


Wy multiply elements

20z-1/2z-234=0

We multiply all the terms by the denominator


20z*2z-234*2z-1=0

Wy multiply elements

40z^2-468z-1=0

a = 40; b = -468; c = -1;
Δ = b2-4ac
Δ = -4682-4·40·(-1)
Δ = 219184
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:

$\sqrt{\Delta}=\sqrt{219184}=\sqrt{16*13699}=\sqrt{16}*\sqrt{13699}=4\sqrt{13699}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-468)-4\sqrt{13699}}{2*40}=\frac{468-4\sqrt{13699}}{80}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-468)+4\sqrt{13699}}{2*40}=\frac{468+4\sqrt{13699}}{80}
$