n+1/2n=9

Solution for n+1/2n=9 equation:

n+1/2n=9

We move all terms to the left:

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


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We multiply all the terms by the denominator


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

Wy multiply elements

2n^2-18n+1=0

a = 2; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·2·1
Δ = 316
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:

$\sqrt{\Delta}=\sqrt{316}=\sqrt{4*79}=\sqrt{4}*\sqrt{79}=2\sqrt{79}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{79}}{2*2}=\frac{18-2\sqrt{79}}{4}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{79}}{2*2}=\frac{18+2\sqrt{79}}{4}
$