1/23n+12=n+14

Solution for 1/23n+12=n+14 equation:

1/23n+12=n+14

We move all terms to the left:

1/23n+12-(n+14)=0


Domain of the equation: 23n!=0

n!=0/23

n!=0

n∈R


We get rid of parentheses

1/23n-n-14+12=0

We multiply all the terms by the denominator


-n*23n-14*23n+12*23n+1=0

Wy multiply elements

-23n^2-322n+276n+1=0

We add all the numbers together, and all the variables

-23n^2-46n+1=0

a = -23; b = -46; c = +1;
Δ = b2-4ac
Δ = -462-4·(-23)·1
Δ = 2208
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:

$\sqrt{\Delta}=\sqrt{2208}=\sqrt{16*138}=\sqrt{16}*\sqrt{138}=4\sqrt{138}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-46)-4\sqrt{138}}{2*-23}=\frac{46-4\sqrt{138}}{-46}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-46)+4\sqrt{138}}{2*-23}=\frac{46+4\sqrt{138}}{-46}
$