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

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

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

We move all terms to the left:

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


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-12x^2-8x-24x+1=0

We add all the numbers together, and all the variables

-12x^2-32x+1=0

a = -12; b = -32; c = +1;
Δ = b2-4ac
Δ = -322-4·(-12)·1
Δ = 1072
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:

$\sqrt{\Delta}=\sqrt{1072}=\sqrt{16*67}=\sqrt{16}*\sqrt{67}=4\sqrt{67}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-32)-4\sqrt{67}}{2*-12}=\frac{32-4\sqrt{67}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-32)+4\sqrt{67}}{2*-12}=\frac{32+4\sqrt{67}}{-24}
$