1/2x-2=5-3x

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

1/2x-2=5-3x

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2-10x-4x+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 = 172
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:

$\sqrt{\Delta}=\sqrt{172}=\sqrt{4*43}=\sqrt{4}*\sqrt{43}=2\sqrt{43}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{43}}{2*6}=\frac{14-2\sqrt{43}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{43}}{2*6}=\frac{14+2\sqrt{43}}{12}
$