1/2x+32+3x-14=31/2x+18

Solution for 1/2x+32+3x-14=31/2x+18 equation:

1/2x+32+3x-14=31/2x+18

We move all terms to the left:

1/2x+32+3x-14-(31/2x+18)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 2x+18)!=0

x∈R


We add all the numbers together, and all the variables

3x+1/2x-(31/2x+18)+18=0

We get rid of parentheses

3x+1/2x-31/2x-18+18=0

We multiply all the terms by the denominator


3x*2x-18*2x+18*2x+1-31=0

We add all the numbers together, and all the variables

3x*2x-18*2x+18*2x-30=0

Wy multiply elements

6x^2-36x+36x-30=0

We add all the numbers together, and all the variables

6x^2-30=0

a = 6; b = 0; c = -30;
Δ = b2-4ac
Δ = 02-4·6·(-30)
Δ = 720
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:

$\sqrt{\Delta}=\sqrt{720}=\sqrt{144*5}=\sqrt{144}*\sqrt{5}=12\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{5}}{2*6}=\frac{0-12\sqrt{5}}{12}
=-\frac{12\sqrt{5}}{12}
=-\sqrt{5}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{5}}{2*6}=\frac{0+12\sqrt{5}}{12}
=\frac{12\sqrt{5}}{12}
=\sqrt{5}
$