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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



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

x∈R


We get rid of parentheses

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

We calculate fractions


(12x+1)/8x^2+(-2x)/8x^2-6-2=0

We add all the numbers together, and all the variables

(12x+1)/8x^2+(-2x)/8x^2-8=0

We multiply all the terms by the denominator


(12x+1)+(-2x)-8*8x^2=0

Wy multiply elements

-64x^2+(12x+1)+(-2x)=0

We get rid of parentheses

-64x^2+12x-2x+1=0

We add all the numbers together, and all the variables

-64x^2+10x+1=0

a = -64; b = 10; c = +1;
Δ = b2-4ac
Δ = 102-4·(-64)·1
Δ = 356
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:

$\sqrt{\Delta}=\sqrt{356}=\sqrt{4*89}=\sqrt{4}*\sqrt{89}=2\sqrt{89}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{89}}{2*-64}=\frac{-10-2\sqrt{89}}{-128}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{89}}{2*-64}=\frac{-10+2\sqrt{89}}{-128}
$