3n-18=1/4n-4

Solution for 3n-18=1/4n-4 equation:

3n-18=1/4n-4

We move all terms to the left:

3n-18-(1/4n-4)=0


Domain of the equation: 4n-4)!=0

n∈R


We get rid of parentheses

3n-1/4n+4-18=0

We multiply all the terms by the denominator


3n*4n+4*4n-18*4n-1=0

Wy multiply elements

12n^2+16n-72n-1=0

We add all the numbers together, and all the variables

12n^2-56n-1=0

a = 12; b = -56; c = -1;
Δ = b2-4ac
Δ = -562-4·12·(-1)
Δ = 3184
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{3184}=\sqrt{16*199}=\sqrt{16}*\sqrt{199}=4\sqrt{199}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-56)-4\sqrt{199}}{2*12}=\frac{56-4\sqrt{199}}{24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-56)+4\sqrt{199}}{2*12}=\frac{56+4\sqrt{199}}{24}
$