1/2n-9=2n-37

Solution for 1/2n-9=2n-37 equation:

1/2n-9=2n-37

We move all terms to the left:

1/2n-9-(2n-37)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-2n+37-9=0

We multiply all the terms by the denominator


-2n*2n+37*2n-9*2n+1=0

Wy multiply elements

-4n^2+74n-18n+1=0

We add all the numbers together, and all the variables

-4n^2+56n+1=0

a = -4; b = 56; c = +1;
Δ = b2-4ac
Δ = 562-4·(-4)·1
Δ = 3152
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:

$\sqrt{\Delta}=\sqrt{3152}=\sqrt{16*197}=\sqrt{16}*\sqrt{197}=4\sqrt{197}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(56)-4\sqrt{197}}{2*-4}=\frac{-56-4\sqrt{197}}{-8}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(56)+4\sqrt{197}}{2*-4}=\frac{-56+4\sqrt{197}}{-8}
$