1/2x-5+3/2x=7/6x+4

Solution for 1/2x-5+3/2x=7/6x+4 equation:

1/2x-5+3/2x=7/6x+4

We move all terms to the left:

1/2x-5+3/2x-(7/6x+4)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 6x+4)!=0

x∈R


We get rid of parentheses

1/2x+3/2x-7/6x-4-5=0

We calculate fractions


(18x+1)/12x^2+(-14x)/12x^2-4-5=0

We add all the numbers together, and all the variables

(18x+1)/12x^2+(-14x)/12x^2-9=0

We multiply all the terms by the denominator


(18x+1)+(-14x)-9*12x^2=0

Wy multiply elements

-108x^2+(18x+1)+(-14x)=0

We get rid of parentheses

-108x^2+18x-14x+1=0

We add all the numbers together, and all the variables

-108x^2+4x+1=0

a = -108; b = 4; c = +1;
Δ = b2-4ac
Δ = 42-4·(-108)·1
Δ = 448
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:

$\sqrt{\Delta}=\sqrt{448}=\sqrt{64*7}=\sqrt{64}*\sqrt{7}=8\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-8\sqrt{7}}{2*-108}=\frac{-4-8\sqrt{7}}{-216}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+8\sqrt{7}}{2*-108}=\frac{-4+8\sqrt{7}}{-216}
$