Step-by-step explanation:
To solve the rational equation (1/3n^2) - ((n^2+6n+5)/3n^2) = 1/3, we need to simplify the equation and then find the values of n that satisfy the equation.
First, let's simplify the equation by combining like terms. We can start by finding a common denominator for the fractions on the right side of the equation:
(1/3n^2) - ((n^2+6n+5)/3n^2) = 1/3
The common denominator for the fractions is 3n^2. To get the first fraction to have a denominator of 3n^2, we can multiply the numerator and denominator by n^2:
(1/3n^2) - ((n^2+6n+5)/3n^2) = 1/3
((1*n^2)/(1*3n^2)) - ((n^2+6n+5)/3n^2) = 1/3
(n^2)/(3n^2) - ((n^2+6n+5)/3n^2) = 1/3
Now, let's simplify further:
(n^2)/(3n^2) - ((n^2+6n+5)/3n^2) = 1/3
(n^2 - (n^2 + 6n + 5))/(3n^2) = 1/3
(n^2 - n^2 - 6n - 5)/(3n^2) = 1/3
(-6n - 5)/(3n^2) = 1/3
Next, let's cross multiply to eliminate the fractions:
(-6n - 5)/(3n^2) = 1/3
3(1/3) = (3n^2)/(-6n - 5)
1 = (3n^2)/(-6n - 5)
Now, let's solve for n by multiplying both sides of the equation by (-6n - 5):
1(-6n - 5) = (3n^2)(-6n - 5)
-6n - 5 = -18n^3 - 15n^2
Now, let's simplify and rearrange the equation:
-6n - 5 = -18n^3 - 15n^2
0 = -18n^3 - 15n^2 + 6n + 5
To solve this cubic equation, we can use synthetic division or factorization methods. However, this equation may not have rational solutions due to the presence of the cubic term.
To check for extraneous solutions, we substitute the possible solutions back into the original equation and see if they satisfy it. However, since we haven't found the values of n yet, we can't check for extraneous solutions at this stage.
In summary, we have simplified the rational equation and obtained a cubic equation. However, we have not yet found the values of n that satisfy the equation, and therefore we cannot check for extraneous solutions
