7/3n-3=5-8n+12

Solution for 7/3n-3=5-8n+12 equation:

7/3n-3=5-8n+12

We move all terms to the left:

7/3n-3-(5-8n+12)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

7/3n-(-8n+17)-3=0

We get rid of parentheses

7/3n+8n-17-3=0

We multiply all the terms by the denominator


8n*3n-17*3n-3*3n+7=0

Wy multiply elements

24n^2-51n-9n+7=0

We add all the numbers together, and all the variables

24n^2-60n+7=0

a = 24; b = -60; c = +7;
Δ = b2-4ac
Δ = -602-4·24·7
Δ = 2928
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:

$\sqrt{\Delta}=\sqrt{2928}=\sqrt{16*183}=\sqrt{16}*\sqrt{183}=4\sqrt{183}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-60)-4\sqrt{183}}{2*24}=\frac{60-4\sqrt{183}}{48}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-60)+4\sqrt{183}}{2*24}=\frac{60+4\sqrt{183}}{48}
$