4x-6=-1/2x+3

Solution for 4x-6=-1/2x+3 equation:

4x-6=-1/2x+3

We move all terms to the left:

4x-6-(-1/2x+3)=0


Domain of the equation: 2x+3)!=0

x∈R


We get rid of parentheses

4x+1/2x-3-6=0

We multiply all the terms by the denominator


4x*2x-3*2x-6*2x+1=0

Wy multiply elements

8x^2-6x-12x+1=0

We add all the numbers together, and all the variables

8x^2-18x+1=0

a = 8; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·8·1
Δ = 292
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:

$\sqrt{\Delta}=\sqrt{292}=\sqrt{4*73}=\sqrt{4}*\sqrt{73}=2\sqrt{73}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{73}}{2*8}=\frac{18-2\sqrt{73}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{73}}{2*8}=\frac{18+2\sqrt{73}}{16}
$