3z-1/2z=10+z

Solution for 3z-1/2z=10+z equation:

3z-1/2z=10+z

We move all terms to the left:

3z-1/2z-(10+z)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


We add all the numbers together, and all the variables

3z-1/2z-(z+10)=0

We get rid of parentheses

3z-1/2z-z-10=0

We multiply all the terms by the denominator


3z*2z-z*2z-10*2z-1=0

Wy multiply elements

6z^2-2z^2-20z-1=0

We add all the numbers together, and all the variables

4z^2-20z-1=0

a = 4; b = -20; c = -1;
Δ = b2-4ac
Δ = -202-4·4·(-1)
Δ = 416
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:

$\sqrt{\Delta}=\sqrt{416}=\sqrt{16*26}=\sqrt{16}*\sqrt{26}=4\sqrt{26}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-4\sqrt{26}}{2*4}=\frac{20-4\sqrt{26}}{8}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+4\sqrt{26}}{2*4}=\frac{20+4\sqrt{26}}{8}
$