3+5n=1/2n+12

Solution for 3+5n=1/2n+12 equation:

3+5n=1/2n+12

We move all terms to the left:

3+5n-(1/2n+12)=0


Domain of the equation: 2n+12)!=0

n∈R


We get rid of parentheses

5n-1/2n-12+3=0

We multiply all the terms by the denominator


5n*2n-12*2n+3*2n-1=0

Wy multiply elements

10n^2-24n+6n-1=0

We add all the numbers together, and all the variables

10n^2-18n-1=0

a = 10; b = -18; c = -1;
Δ = b2-4ac
Δ = -182-4·10·(-1)
Δ = 364
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:

$\sqrt{\Delta}=\sqrt{364}=\sqrt{4*91}=\sqrt{4}*\sqrt{91}=2\sqrt{91}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{91}}{2*10}=\frac{18-2\sqrt{91}}{20}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{91}}{2*10}=\frac{18+2\sqrt{91}}{20}
$