1/515n+5=8n+-9

Solution for 1/515n+5=8n+-9 equation:

1/515n+5=8n+-9

We move all terms to the left:

1/515n+5-(8n+-9)=0


Domain of the equation: 515n!=0

n!=0/515

n!=0

n∈R


We add all the numbers together, and all the variables

1/515n-(8n-9)+5=0

We get rid of parentheses

1/515n-8n+9+5=0

We multiply all the terms by the denominator


-8n*515n+9*515n+5*515n+1=0

Wy multiply elements

-4120n^2+4635n+2575n+1=0

We add all the numbers together, and all the variables

-4120n^2+7210n+1=0

a = -4120; b = 7210; c = +1;
Δ = b2-4ac
Δ = 72102-4·(-4120)·1
Δ = 52000580
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:

$\sqrt{\Delta}=\sqrt{52000580}=\sqrt{4*13000145}=\sqrt{4}*\sqrt{13000145}=2\sqrt{13000145}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(7210)-2\sqrt{13000145}}{2*-4120}=\frac{-7210-2\sqrt{13000145}}{-8240}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(7210)+2\sqrt{13000145}}{2*-4120}=\frac{-7210+2\sqrt{13000145}}{-8240}
$