0.2n+13=1/3n-14.5

Solution for 0.2n+13=1/3n-14.5 equation:

0.2n+13=1/3n-14.5

We move all terms to the left:

0.2n+13-(1/3n-14.5)=0


Domain of the equation: 3n-14.5)!=0

n∈R


We get rid of parentheses

0.2n-1/3n+14.5+13=0

We multiply all the terms by the denominator


(0.2n)*3n+(14.5)*3n+13*3n-1=0

We add all the numbers together, and all the variables

(+0.2n)*3n+(14.5)*3n+13*3n-1=0

We multiply parentheses

0n^2+43.5n+13*3n-1=0

Wy multiply elements

0n^2+43.5n+39n-1=0

We add all the numbers together, and all the variables

n^2+82.5n-1=0

a = 1; b = 82.5; c = -1;
Δ = b2-4ac
Δ = 82.52-4·1·(-1)
Δ = 6810.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(82.5)-\sqrt{6810.25}}{2*1}=\frac{-82.5-\sqrt{6810.25}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(82.5)+\sqrt{6810.25}}{2*1}=\frac{-82.5+\sqrt{6810.25}}{2}
$