3x+4=1/2x-2

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

3x+4=1/2x-2

We move all terms to the left:

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


Domain of the equation: 2x-2)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2+4x+8x-1=0

We add all the numbers together, and all the variables

6x^2+12x-1=0

a = 6; b = 12; c = -1;
Δ = b2-4ac
Δ = 122-4·6·(-1)
Δ = 168
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:

$\sqrt{\Delta}=\sqrt{168}=\sqrt{4*42}=\sqrt{4}*\sqrt{42}=2\sqrt{42}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(12)-2\sqrt{42}}{2*6}=\frac{-12-2\sqrt{42}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(12)+2\sqrt{42}}{2*6}=\frac{-12+2\sqrt{42}}{12}
$