5/(n^3=5n^2)=4/(n+5)+1/n^2
1. I assume you are solving
2. I assume that the = should be + (since they are on the same key)
so
5/(n^3+5n^2)=4/(n+5)+1/n^2
5/(n^2(n+5)) = 4/(n+5) = 1/n^2
multiply both sides by n^2(n+5)
5 = 4n^2 + n+5
4n^2 + n = 0
n(4n+1) = 0
n = 0 or n = -1/4
but n cannot be zero because of the original last term
so
n = -1/4
thank you sooo much :)
To solve the given equation, let's first simplify each side.
Starting with the left side of the equation, we have:
5 / (n^3 - 5n^2)
The denominator n^3 - 5n^2 can be factored out as n^2(n - 5). Therefore, the left side becomes:
5 / (n^2(n - 5))
Moving on to the right side:
4 / (n +5) + 1 / n^2
To simplify this further, we need a common denominator for the two fractions on the right side. The common denominator can be obtained by multiplying n^2 to (n + 5), resulting in:
4n^2 / (n^2(n + 5)) + 1 / n^2
Now, we can add these two fractions since they have the same denominator:
(4n^2 + 1) / (n^2(n + 5))
Now that we've simplified both sides of the equation, we can set them equal to each other:
5 / (n^2(n - 5)) = (4n^2 + 1) / (n^2(n + 5))
To continue solving this, we can cross multiply by multiplying the numerator of the left fraction with the denominator of the right fraction, and vice versa:
5 * (n^2 * (n + 5)) = (4n^2 + 1) * (n^2 * (n - 5))
Expanding and simplifying these expressions, we get:
5n^2 * (n + 5) = (4n^2 + 1) * (n^2 * (n - 5))
5n^3 + 25n^2 = (4n^2 + 1) * (n^3 - 5n^2)
Now, distribute the terms on the right side:
5n^3 + 25n^2 = 4n^5 - 20n^4 + n^3 - 5n^2
Grouping like terms:
0 = 4n^5 - 20n^4 - 4n^3 + 30n^2
Now, we have a fifth-degree polynomial equation. Unfortunately, there's no straightforward algebraic method to solve equations of this degree. You would typically need to use numerical methods or approximation techniques to find the solutions.