1/2z+9=-2/3z

Solution for 1/2z+9=-2/3z equation:

1/2z+9=-2/3z

We move all terms to the left:

1/2z+9-(-2/3z)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R



Domain of the equation: 3z)!=0

z!=0/1

z!=0

z∈R


We get rid of parentheses

1/2z+2/3z+9=0

We calculate fractions


3z/6z^2+4z/6z^2+9=0

We multiply all the terms by the denominator


3z+4z+9*6z^2=0

We add all the numbers together, and all the variables

7z+9*6z^2=0

Wy multiply elements

54z^2+7z=0

a = 54; b = 7; c = 0;
Δ = b2-4ac
Δ = 72-4·54·0
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{-(7)-7}{2*54}=\frac{-14}{108}
=-7/54
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(7)+7}{2*54}=\frac{0}{108}
=0
$