-1/2n+1=7n-14

Solution for -1/2n+1=7n-14 equation:

-1/2n+1=7n-14

We move all terms to the left:

-1/2n+1-(7n-14)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

-1/2n-7n+14+1=0

We multiply all the terms by the denominator


-7n*2n+14*2n+1*2n-1=0

Wy multiply elements

-14n^2+28n+2n-1=0

We add all the numbers together, and all the variables

-14n^2+30n-1=0

a = -14; b = 30; c = -1;
Δ = b2-4ac
Δ = 302-4·(-14)·(-1)
Δ = 844
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:

$\sqrt{\Delta}=\sqrt{844}=\sqrt{4*211}=\sqrt{4}*\sqrt{211}=2\sqrt{211}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{211}}{2*-14}=\frac{-30-2\sqrt{211}}{-28}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{211}}{2*-14}=\frac{-30+2\sqrt{211}}{-28}
$