3+1/2n=17-3n

Solution for 3+1/2n=17-3n equation:

3+1/2n=17-3n

We move all terms to the left:

3+1/2n-(17-3n)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

1/2n-(-3n+17)+3=0

We get rid of parentheses

1/2n+3n-17+3=0

We multiply all the terms by the denominator


3n*2n-17*2n+3*2n+1=0

Wy multiply elements

6n^2-34n+6n+1=0

We add all the numbers together, and all the variables

6n^2-28n+1=0

a = 6; b = -28; c = +1;
Δ = b2-4ac
Δ = -282-4·6·1
Δ = 760
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:

$\sqrt{\Delta}=\sqrt{760}=\sqrt{4*190}=\sqrt{4}*\sqrt{190}=2\sqrt{190}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-2\sqrt{190}}{2*6}=\frac{28-2\sqrt{190}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+2\sqrt{190}}{2*6}=\frac{28+2\sqrt{190}}{12}
$