n+1=1/3n-7

Solution for n+1=1/3n-7 equation:

n+1=1/3n-7

We move all terms to the left:

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


Domain of the equation: 3n-7)!=0

n∈R


We get rid of parentheses

n-1/3n+7+1=0

We multiply all the terms by the denominator


n*3n+7*3n+1*3n-1=0

Wy multiply elements

3n^2+21n+3n-1=0

We add all the numbers together, and all the variables

3n^2+24n-1=0

a = 3; b = 24; c = -1;
Δ = b2-4ac
Δ = 242-4·3·(-1)
Δ = 588
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:

$\sqrt{\Delta}=\sqrt{588}=\sqrt{196*3}=\sqrt{196}*\sqrt{3}=14\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-14\sqrt{3}}{2*3}=\frac{-24-14\sqrt{3}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+14\sqrt{3}}{2*3}=\frac{-24+14\sqrt{3}}{6}
$