7n+12=1/24n+24

Solution for 7n+12=1/24n+24 equation:

7n+12=1/24n+24

We move all terms to the left:

7n+12-(1/24n+24)=0


Domain of the equation: 24n+24)!=0

n∈R


We get rid of parentheses

7n-1/24n-24+12=0

We multiply all the terms by the denominator


7n*24n-24*24n+12*24n-1=0

Wy multiply elements

168n^2-576n+288n-1=0

We add all the numbers together, and all the variables

168n^2-288n-1=0

a = 168; b = -288; c = -1;
Δ = b2-4ac
Δ = -2882-4·168·(-1)
Δ = 83616
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:

$\sqrt{\Delta}=\sqrt{83616}=\sqrt{16*5226}=\sqrt{16}*\sqrt{5226}=4\sqrt{5226}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-288)-4\sqrt{5226}}{2*168}=\frac{288-4\sqrt{5226}}{336}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-288)+4\sqrt{5226}}{2*168}=\frac{288+4\sqrt{5226}}{336}
$