1/2n-9=4n-5

Solution for 1/2n-9=4n-5 equation:

1/2n-9=4n-5

We move all terms to the left:

1/2n-9-(4n-5)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-4n+5-9=0

We multiply all the terms by the denominator


-4n*2n+5*2n-9*2n+1=0

Wy multiply elements

-8n^2+10n-18n+1=0

We add all the numbers together, and all the variables

-8n^2-8n+1=0

a = -8; b = -8; c = +1;
Δ = b2-4ac
Δ = -82-4·(-8)·1
Δ = 96
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:

$\sqrt{\Delta}=\sqrt{96}=\sqrt{16*6}=\sqrt{16}*\sqrt{6}=4\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{6}}{2*-8}=\frac{8-4\sqrt{6}}{-16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{6}}{2*-8}=\frac{8+4\sqrt{6}}{-16}
$