1/2+1/6q=2/3q-3

Solution for 1/2+1/6q=2/3q-3 equation:

1/2+1/6q=2/3q-3

We move all terms to the left:

1/2+1/6q-(2/3q-3)=0


Domain of the equation: 6q!=0

q!=0/6

q!=0

q∈R



Domain of the equation: 3q-3)!=0

q∈R


We get rid of parentheses

1/6q-2/3q+3+1/2=0

We calculate fractions


54q^2/72q^2+12q/72q^2+(-48q)/72q^2+3=0

We multiply all the terms by the denominator


54q^2+12q+(-48q)+3*72q^2=0

Wy multiply elements

54q^2+216q^2+12q+(-48q)=0

We get rid of parentheses

54q^2+216q^2+12q-48q=0

We add all the numbers together, and all the variables

270q^2-36q=0

a = 270; b = -36; c = 0;
Δ = b2-4ac
Δ = -362-4·270·0
Δ = 1296
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1296}=36$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-36}{2*270}=\frac{0}{540}
=0
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+36}{2*270}=\frac{72}{540}
=2/15
$