5/7n=10n

Solution for 5/7n=10n equation:

5/7n=10n

We move all terms to the left:

5/7n-(10n)=0


Domain of the equation: 7n!=0

n!=0/7

n!=0

n∈R


We add all the numbers together, and all the variables

-10n+5/7n=0

We multiply all the terms by the denominator


-10n*7n+5=0

Wy multiply elements

-70n^2+5=0

a = -70; b = 0; c = +5;
Δ = b2-4ac
Δ = 02-4·(-70)·5
Δ = 1400
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:

$\sqrt{\Delta}=\sqrt{1400}=\sqrt{100*14}=\sqrt{100}*\sqrt{14}=10\sqrt{14}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-10\sqrt{14}}{2*-70}=\frac{0-10\sqrt{14}}{-140}
=-\frac{10\sqrt{14}}{-140}
=-\frac{\sqrt{14}}{-14}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+10\sqrt{14}}{2*-70}=\frac{0+10\sqrt{14}}{-140}
=\frac{10\sqrt{14}}{-140}
=\frac{\sqrt{14}}{-14}
$