31/2x+2=x+3

Solution for 31/2x+2=x+3 equation:

31/2x+2=x+3

We move all terms to the left:

31/2x+2-(x+3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

31/2x-x-3+2=0

We multiply all the terms by the denominator


-x*2x-3*2x+2*2x+31=0

Wy multiply elements

-2x^2-6x+4x+31=0

We add all the numbers together, and all the variables

-2x^2-2x+31=0

a = -2; b = -2; c = +31;
Δ = b2-4ac
Δ = -22-4·(-2)·31
Δ = 252
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:

$\sqrt{\Delta}=\sqrt{252}=\sqrt{36*7}=\sqrt{36}*\sqrt{7}=6\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-6\sqrt{7}}{2*-2}=\frac{2-6\sqrt{7}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+6\sqrt{7}}{2*-2}=\frac{2+6\sqrt{7}}{-4}
$