1/2x+23=3x-2

Solution for 1/2x+23=3x-2 equation:

1/2x+23=3x-2

We move all terms to the left:

1/2x+23-(3x-2)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+2+23=0

We multiply all the terms by the denominator


-3x*2x+2*2x+23*2x+1=0

Wy multiply elements

-6x^2+4x+46x+1=0

We add all the numbers together, and all the variables

-6x^2+50x+1=0

a = -6; b = 50; c = +1;
Δ = b2-4ac
Δ = 502-4·(-6)·1
Δ = 2524
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:

$\sqrt{\Delta}=\sqrt{2524}=\sqrt{4*631}=\sqrt{4}*\sqrt{631}=2\sqrt{631}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(50)-2\sqrt{631}}{2*-6}=\frac{-50-2\sqrt{631}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(50)+2\sqrt{631}}{2*-6}=\frac{-50+2\sqrt{631}}{-12}
$