.5w+3=2/3w-5

Solution for .5w+3=2/3w-5 equation:

.5w+3=2/3w-5

We move all terms to the left:

.5w+3-(2/3w-5)=0


Domain of the equation: 3w-5)!=0

w∈R


We get rid of parentheses

.5w-2/3w+5+3=0

We multiply all the terms by the denominator


(.5w)*3w+5*3w+3*3w-2=0

We add all the numbers together, and all the variables

(+.5w)*3w+5*3w+3*3w-2=0

We multiply parentheses

3w^2+5*3w+3*3w-2=0

Wy multiply elements

3w^2+15w+9w-2=0

We add all the numbers together, and all the variables

3w^2+24w-2=0

a = 3; b = 24; c = -2;
Δ = b2-4ac
Δ = 242-4·3·(-2)
Δ = 600
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{600}=\sqrt{100*6}=\sqrt{100}*\sqrt{6}=10\sqrt{6}$
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-10\sqrt{6}}{2*3}=\frac{-24-10\sqrt{6}}{6}
$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+10\sqrt{6}}{2*3}=\frac{-24+10\sqrt{6}}{6}
$