(3/4)n=61

Solution for (3/4)n=61 equation:

(3/4)n=61

We move all terms to the left:

(3/4)n-(61)=0


Domain of the equation: 4)n!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

(+3/4)n-61=0

We multiply parentheses

3n^2-61=0

a = 3; b = 0; c = -61;
Δ = b2-4ac
Δ = 02-4·3·(-61)
Δ = 732
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:

$\sqrt{\Delta}=\sqrt{732}=\sqrt{4*183}=\sqrt{4}*\sqrt{183}=2\sqrt{183}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{183}}{2*3}=\frac{0-2\sqrt{183}}{6}
=-\frac{2\sqrt{183}}{6}
=-\frac{\sqrt{183}}{3}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{183}}{2*3}=\frac{0+2\sqrt{183}}{6}
=\frac{2\sqrt{183}}{6}
=\frac{\sqrt{183}}{3}
$