1/2x+3=3x-4

Solution for 1/2x+3=3x-4 equation:

1/2x+3=3x-4

We move all terms to the left:

1/2x+3-(3x-4)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+4+3=0

We multiply all the terms by the denominator


-3x*2x+4*2x+3*2x+1=0

Wy multiply elements

-6x^2+8x+6x+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 = 220
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:

$\sqrt{\Delta}=\sqrt{220}=\sqrt{4*55}=\sqrt{4}*\sqrt{55}=2\sqrt{55}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{55}}{2*-6}=\frac{-14-2\sqrt{55}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{55}}{2*-6}=\frac{-14+2\sqrt{55}}{-12}
$