3x-5=1/2x+2

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

3x-5=1/2x+2

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2-4x-10x-1=0

We add all the numbers together, and all the variables

6x^2-14x-1=0

a = 6; b = -14; c = -1;
Δ = b2-4ac
Δ = -142-4·6·(-1)
Δ = 220
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{220}=\sqrt{4*55}=\sqrt{4}*\sqrt{55}=2\sqrt{55}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{55}}{2*6}=\frac{14-2\sqrt{55}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{55}}{2*6}=\frac{14+2\sqrt{55}}{12}
$