4n=3/2n+9

Solution for 4n=3/2n+9 equation:

4n=3/2n+9

We move all terms to the left:

4n-(3/2n+9)=0


Domain of the equation: 2n+9)!=0

n∈R


We get rid of parentheses

4n-3/2n-9=0

We multiply all the terms by the denominator


4n*2n-9*2n-3=0

Wy multiply elements

8n^2-18n-3=0

a = 8; b = -18; c = -3;
Δ = b2-4ac
Δ = -182-4·8·(-3)
Δ = 420
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:

$\sqrt{\Delta}=\sqrt{420}=\sqrt{4*105}=\sqrt{4}*\sqrt{105}=2\sqrt{105}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{105}}{2*8}=\frac{18-2\sqrt{105}}{16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{105}}{2*8}=\frac{18+2\sqrt{105}}{16}
$