5/6z+1/2=2/3z+4

Solution for 5/6z+1/2=2/3z+4 equation:

5/6z+1/2=2/3z+4

We move all terms to the left:

5/6z+1/2-(2/3z+4)=0


Domain of the equation: 6z!=0

z!=0/6

z!=0

z∈R



Domain of the equation: 3z+4)!=0

z∈R


We get rid of parentheses

5/6z-2/3z-4+1/2=0

We calculate fractions


54z^2/72z^2+60z/72z^2+(-48z)/72z^2-4=0

We multiply all the terms by the denominator


54z^2+60z+(-48z)-4*72z^2=0

Wy multiply elements

54z^2-288z^2+60z+(-48z)=0

We get rid of parentheses

54z^2-288z^2+60z-48z=0

We add all the numbers together, and all the variables

-234z^2+12z=0

a = -234; b = 12; c = 0;
Δ = b2-4ac
Δ = 122-4·(-234)·0
Δ = 144
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{144}=12$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(12)-12}{2*-234}=\frac{-24}{-468}
=2/39
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(12)+12}{2*-234}=\frac{0}{-468}
=0
$