1/2n+5=3n-5

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

1/2n+5=3n-5

We move all terms to the left:

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


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-6n^2+20n+1=0

a = -6; b = 20; c = +1;
Δ = b2-4ac
Δ = 202-4·(-6)·1
Δ = 424
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:

$\sqrt{\Delta}=\sqrt{424}=\sqrt{4*106}=\sqrt{4}*\sqrt{106}=2\sqrt{106}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-2\sqrt{106}}{2*-6}=\frac{-20-2\sqrt{106}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+2\sqrt{106}}{2*-6}=\frac{-20+2\sqrt{106}}{-12}
$