3/4x+3x-1/4x=1.5x-6

Solution for 3/4x+3x-1/4x=1.5x-6 equation:

3/4x+3x-1/4x=1.5x-6

We move all terms to the left:

3/4x+3x-1/4x-(1.5x-6)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We add all the numbers together, and all the variables

3x+3/4x-1/4x-(1.5x-6)=0

We get rid of parentheses

3x+3/4x-1/4x-1.5x+6=0

We multiply all the terms by the denominator


3x*4x-(1.5x)*4x+6*4x+3-1=0

We add all the numbers together, and all the variables

3x*4x-(+1.5x)*4x+6*4x+3-1=0

We add all the numbers together, and all the variables

3x*4x-(+1.5x)*4x+6*4x+2=0

We multiply parentheses

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

Wy multiply elements

-4x^2+12x^2+24x+2=0

We add all the numbers together, and all the variables

8x^2+24x+2=0

a = 8; b = 24; c = +2;
Δ = b2-4ac
Δ = 242-4·8·2
Δ = 512
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:

$\sqrt{\Delta}=\sqrt{512}=\sqrt{256*2}=\sqrt{256}*\sqrt{2}=16\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-16\sqrt{2}}{2*8}=\frac{-24-16\sqrt{2}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+16\sqrt{2}}{2*8}=\frac{-24+16\sqrt{2}}{16}
$