1/4k.k.k.k.k.k.k-3k.k.k=2

Solution for 1/4k.k.k.k.k.k.k-3k.k.k=2 equation:

1/4k.k.k.k.k.k.k-3k.k.k=2

We move all terms to the left:

1/4k.k.k.k.k.k.k-3k.k.k-(2)=0


Domain of the equation: 4k.k.k.k.k.k.k!=0

k!=0/1

k!=0

k∈R


We multiply all the terms by the denominator


-(3k.k.k)*4k.k.k.k.k.k.k-2*4k.k.k.k.k.k.k+1=0

We add all the numbers together, and all the variables

-(+3k.k.k)*4k.k.k.k.k.k.k-2*4k.k.k.k.k.k.k+1=0

We multiply parentheses

-12k^2-2*4k.k.k.k.k.k.k+1=0

Wy multiply elements

-12k^2-8k+1=0

a = -12; b = -8; c = +1;
Δ = b2-4ac
Δ = -82-4·(-12)·1
Δ = 112
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{112}=\sqrt{16*7}=\sqrt{16}*\sqrt{7}=4\sqrt{7}$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{7}}{2*-12}=\frac{8-4\sqrt{7}}{-24}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{7}}{2*-12}=\frac{8+4\sqrt{7}}{-24}
$