Solve by Factoring 5/(n^3+5n^2)=4/(n+5)+1/(n^2)

$\frac{5}{n^{3} + 5 n^{2}} = \frac{4}{n + 5} + \frac{1}{n^{2}}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{4}{n + 5}$ from both sides of the equation.

$\frac{5}{n^{3} + 5 n^{2}} - \frac{4}{n + 5} = \frac{1}{n^{2}}$

Subtract $\frac{1}{n^{2}}$ from both sides of the equation.

$\frac{5}{n^{3} + 5 n^{2}} - \frac{4}{n + 5} - \frac{1}{n^{2}} = 0$

Step 2

Simplify $\frac{5}{n^{3} + 5 n^{2}} - \frac{4}{n + 5} - \frac{1}{n^{2}}$ .

Factor $n^{2}$ out of $n^{3} + 5 n^{2}$ .

Factor $n^{2}$ out of $n^{3}$ .

$\frac{5}{n^{2} n + 5 n^{2}} - \frac{4}{n + 5} - \frac{1}{n^{2}} = 0$

Factor $n^{2}$ out of $5 n^{2}$ .

$\frac{5}{n^{2} n + n^{2} \cdot 5} - \frac{4}{n + 5} - \frac{1}{n^{2}} = 0$

Factor $n^{2}$ out of $n^{2} n + n^{2} \cdot 5$ .

$\frac{5}{n^{2} (n + 5)} - \frac{4}{n + 5} - \frac{1}{n^{2}} = 0$

Find the common denominator.

Multiply $\frac{4}{n + 5}$ by $\frac{n^{2}}{n^{2}}$ .

$\frac{5}{n^{2} (n + 5)} - (\frac{4}{n + 5} \cdot \frac{n^{2}}{n^{2}}) - \frac{1}{n^{2}} = 0$

Multiply $\frac{4}{n + 5}$ by $\frac{n^{2}}{n^{2}}$ .

$\frac{5}{n^{2} (n + 5)} - \frac{4 n^{2}}{(n + 5) n^{2}} - \frac{1}{n^{2}} = 0$

Multiply $\frac{1}{n^{2}}$ by $\frac{n + 5}{n + 5}$ .

$\frac{5}{n^{2} (n + 5)} - \frac{4 n^{2}}{(n + 5) n^{2}} - (\frac{1}{n^{2}} \cdot \frac{n + 5}{n + 5}) = 0$

Multiply $\frac{1}{n^{2}}$ by $\frac{n + 5}{n + 5}$ .

$\frac{5}{n^{2} (n + 5)} - \frac{4 n^{2}}{(n + 5) n^{2}} - \frac{n + 5}{n^{2} (n + 5)} = 0$

Combine the numerators over the common denominator.

$\frac{5 - 4 n^{2} - (n + 5)}{n^{2} (n + 5)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{5 - 4 n^{2} - n - 1 \cdot 5}{n^{2} (n + 5)} = 0$

Multiply $- 1$ by $5$ .

$\frac{5 - 4 n^{2} - n - 5}{n^{2} (n + 5)} = 0$

Combine the opposite terms in $5 - 4 n^{2} - n - 5$ .

Subtract $5$ from $5$ .

$\frac{- 4 n^{2} - n + 0}{n^{2} (n + 5)} = 0$

Add $- 4 n^{2} - n$ and $0$ .

$\frac{- 4 n^{2} - n}{n^{2} (n + 5)} = 0$

Factor $n$ out of $- 4 n^{2} - n$ .

Factor $n$ out of $- 4 n^{2}$ .

$\frac{n (- 4 n) - n}{n^{2} (n + 5)} = 0$

Factor $n$ out of $- n$ .

$\frac{n (- 4 n) + n \cdot - 1}{n^{2} (n + 5)} = 0$

Factor $n$ out of $n (- 4 n) + n \cdot - 1$ .

$\frac{n (- 4 n - 1)}{n^{2} (n + 5)} = 0$

Cancel the common factors.

Factor $n$ out of $n^{2} (n + 5)$ .

$\frac{n (- 4 n - 1)}{n (n (n + 5))} = 0$

Cancel the common factor.

$\frac{n (- 4 n - 1)}{n (n (n + 5))} = 0$

Rewrite the expression.

$\frac{- 4 n - 1}{n (n + 5)} = 0$

Factor $- 1$ out of $- 4 n$ .

$\frac{- (4 n) - 1}{n (n + 5)} = 0$

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- (4 n) - 1 (1)}{n (n + 5)} = 0$

Factor $- 1$ out of $- (4 n) - 1 (1)$ .

$\frac{- (4 n + 1)}{n (n + 5)} = 0$

Rewrite $- (4 n + 1)$ as $- 1 (4 n + 1)$ .

$\frac{- 1 (4 n + 1)}{n (n + 5)} = 0$

Move the negative in front of the fraction.

$- \frac{4 n + 1}{n (n + 5)} = 0$

Step 3

Set the numerator equal to zero.

$4 n + 1 = 0$

Step 4

Solve the equation for $n$ .

Subtract $1$ from both sides of the equation.

$4 n = - 1$

Divide each term in $4 n = - 1$ by $4$ and simplify.

Divide each term in $4 n = - 1$ by $4$ .

$\frac{4 n}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 n}{4} = \frac{- 1}{4}$

Divide $n$ by $1$ .

$n = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$n = - \frac{1}{4}$

Solve by Factoring 5/(n^3+5n^2)=4/(n+5)+1/(n^2)

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