.23n=2/783n-5

Solution for .23n=2/783n-5 equation:

.23n=2/783n-5

We move all terms to the left:

.23n-(2/783n-5)=0


Domain of the equation: 783n-5)!=0

n∈R


We get rid of parentheses

.23n-2/783n+5=0

We multiply all the terms by the denominator


(.23n)*783n+5*783n-2=0

We add all the numbers together, and all the variables

(+.23n)*783n+5*783n-2=0

We multiply parentheses

783n^2+5*783n-2=0

Wy multiply elements

783n^2+3915n-2=0

a = 783; b = 3915; c = -2;
Δ = b2-4ac
Δ = 39152-4·783·(-2)
Δ = 15333489
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:

$\sqrt{\Delta}=\sqrt{15333489}=\sqrt{9*1703721}=\sqrt{9}*\sqrt{1703721}=3\sqrt{1703721}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3915)-3\sqrt{1703721}}{2*783}=\frac{-3915-3\sqrt{1703721}}{1566}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3915)+3\sqrt{1703721}}{2*783}=\frac{-3915+3\sqrt{1703721}}{1566}
$