9-1/2n=5n-4

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

9-1/2n=5n-4

We move all terms to the left:

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


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-10n^2+26n-1=0

a = -10; b = 26; c = -1;
Δ = b2-4ac
Δ = 262-4·(-10)·(-1)
Δ = 636
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:

$\sqrt{\Delta}=\sqrt{636}=\sqrt{4*159}=\sqrt{4}*\sqrt{159}=2\sqrt{159}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(26)-2\sqrt{159}}{2*-10}=\frac{-26-2\sqrt{159}}{-20}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(26)+2\sqrt{159}}{2*-10}=\frac{-26+2\sqrt{159}}{-20}
$