3/x+3-1/x-2=5/2x+6

Solution for 3/x+3-1/x-2=5/2x+6 equation:

3/x+3-1/x-2=5/2x+6

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R



Domain of the equation: 2x+6)!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

3/x-1/x-5/2x-6+1=0

We calculate fractions


(-2x+3)/2x^2+(-5x)/2x^2-6+1=0

We add all the numbers together, and all the variables

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

We multiply all the terms by the denominator


(-2x+3)+(-5x)-5*2x^2=0

Wy multiply elements

-10x^2+(-2x+3)+(-5x)=0

We get rid of parentheses

-10x^2-2x-5x+3=0

We add all the numbers together, and all the variables

-10x^2-7x+3=0

a = -10; b = -7; c = +3;
Δ = b2-4ac
Δ = -72-4·(-10)·3
Δ = 169
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{169}=13$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-13}{2*-10}=\frac{-6}{-20}
=3/10
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+13}{2*-10}=\frac{20}{-20}
=-1
$