1-1/2n=3n+16

Solution for 1-1/2n=3n+16 equation:

1-1/2n=3n+16

We move all terms to the left:

1-1/2n-(3n+16)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

-1/2n-3n-16+1=0

We multiply all the terms by the denominator


-3n*2n-16*2n+1*2n-1=0

Wy multiply elements

-6n^2-32n+2n-1=0

We add all the numbers together, and all the variables

-6n^2-30n-1=0

a = -6; b = -30; c = -1;
Δ = b2-4ac
Δ = -302-4·(-6)·(-1)
Δ = 876
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:

$\sqrt{\Delta}=\sqrt{876}=\sqrt{4*219}=\sqrt{4}*\sqrt{219}=2\sqrt{219}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{219}}{2*-6}=\frac{30-2\sqrt{219}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{219}}{2*-6}=\frac{30+2\sqrt{219}}{-12}
$