7/6n-n=414

Solution for 7/6n-n=414 equation:

7/6n-n=414

We move all terms to the left:

7/6n-n-(414)=0


Domain of the equation: 6n!=0

n!=0/6

n!=0

n∈R


We add all the numbers together, and all the variables

-1n+7/6n-414=0

We multiply all the terms by the denominator


-1n*6n-414*6n+7=0

Wy multiply elements

-6n^2-2484n+7=0

a = -6; b = -2484; c = +7;
Δ = b2-4ac
Δ = -24842-4·(-6)·7
Δ = 6170424
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:

$\sqrt{\Delta}=\sqrt{6170424}=\sqrt{4*1542606}=\sqrt{4}*\sqrt{1542606}=2\sqrt{1542606}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2484)-2\sqrt{1542606}}{2*-6}=\frac{2484-2\sqrt{1542606}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2484)+2\sqrt{1542606}}{2*-6}=\frac{2484+2\sqrt{1542606}}{-12}
$