2(1/4n-7)=1/3n+4

Solution for 2(1/4n-7)=1/3n+4 equation:

2(1/4n-7)=1/3n+4

We move all terms to the left:

2(1/4n-7)-(1/3n+4)=0


Domain of the equation: 4n-7)!=0

n∈R



Domain of the equation: 3n+4)!=0

n∈R


We multiply parentheses

2n-(1/3n+4)-14=0

We get rid of parentheses

2n-1/3n-4-14=0

We multiply all the terms by the denominator


2n*3n-4*3n-14*3n-1=0

Wy multiply elements

6n^2-12n-42n-1=0

We add all the numbers together, and all the variables

6n^2-54n-1=0

a = 6; b = -54; c = -1;
Δ = b2-4ac
Δ = -542-4·6·(-1)
Δ = 2940
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:

$\sqrt{\Delta}=\sqrt{2940}=\sqrt{196*15}=\sqrt{196}*\sqrt{15}=14\sqrt{15}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-14\sqrt{15}}{2*6}=\frac{54-14\sqrt{15}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+14\sqrt{15}}{2*6}=\frac{54+14\sqrt{15}}{12}
$