X-2=1/2x+3

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

X-2=1/2X+3

We move all terms to the left:

X-2-(1/2X+3)=0


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

X∈R


We get rid of parentheses

X-1/2X-3-2=0

We multiply all the terms by the denominator


X*2X-3*2X-2*2X-1=0

Wy multiply elements

2X^2-6X-4X-1=0

We add all the numbers together, and all the variables

2X^2-10X-1=0

a = 2; b = -10; c = -1;
Δ = b2-4ac
Δ = -102-4·2·(-1)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-6\sqrt{3}}{2*2}=\frac{10-6\sqrt{3}}{4}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+6\sqrt{3}}{2*2}=\frac{10+6\sqrt{3}}{4}
$