3/z+3=5z-2-3z

Solution for 3/z+3=5z-2-3z equation:

3/z+3=5z-2-3z

We move all terms to the left:

3/z+3-(5z-2-3z)=0


Domain of the equation: z!=0

z∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

3/z-2z+2+3=0

We multiply all the terms by the denominator


-2z*z+2*z+3*z+3=0

We add all the numbers together, and all the variables

5z-2z*z+3=0

Wy multiply elements

-2z^2+5z+3=0

a = -2; b = 5; c = +3;
Δ = b2-4ac
Δ = 52-4·(-2)·3
Δ = 49
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{49}=7$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(5)-7}{2*-2}=\frac{-12}{-4}
=+3
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+7}{2*-2}=\frac{2}{-4}
=-1/2
$