-1/2x-5=2/3x+2

Solution for -1/2x-5=2/3x+2 equation:

-1/2x-5=2/3x+2

We move all terms to the left:

-1/2x-5-(2/3x+2)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 3x+2)!=0

x∈R


We get rid of parentheses

-1/2x-2/3x-2-5=0

We calculate fractions


(-3x)/6x^2+(-4x)/6x^2-2-5=0

We add all the numbers together, and all the variables

(-3x)/6x^2+(-4x)/6x^2-7=0

We multiply all the terms by the denominator


(-3x)+(-4x)-7*6x^2=0

Wy multiply elements

-42x^2+(-3x)+(-4x)=0

We get rid of parentheses

-42x^2-3x-4x=0

We add all the numbers together, and all the variables

-42x^2-7x=0

a = -42; b = -7; c = 0;
Δ = b2-4ac
Δ = -72-4·(-42)·0
Δ = 49
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{49}=7$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-7}{2*-42}=\frac{0}{-84}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+7}{2*-42}=\frac{14}{-84}
=-1/6
$