Solve (y + 5)/(y ^ 2 + 6y + 9) + 6/(y ^ 2 + 9) - (16y)/(8y ^ 2 - 24y) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{y+5}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{16y}{8y^{2}-24y}$$

Evaluate

$-\frac{y^{4}+4y^{3}+24y^{2}+144y+459}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{16y}{8y^{2}-24y}$.

$$\frac{y+5}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{16y}{8y\left(y-3\right)}$$

Cancel out $8y$ in both numerator and denominator.

$$\frac{y+5}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{2}{y-3}$$

Factor $y^{2}+6y+9$.

$$\frac{y+5}{\left(y+3\right)^{2}}+\frac{6}{y^{2}+9}-\frac{2}{y-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)^{2}$ and $y^{2}+9$ is $\left(y+3\right)^{2}\left(y^{2}+9\right)$. Multiply $\frac{y+5}{\left(y+3\right)^{2}}$ times $\frac{y^{2}+9}{y^{2}+9}$. Multiply $\frac{6}{y^{2}+9}$ times $\frac{\left(y+3\right)^{2}}{\left(y+3\right)^{2}}$.

$$\frac{\left(y+5\right)\left(y^{2}+9\right)}{\left(y+3\right)^{2}\left(y^{2}+9\right)}+\frac{6\left(y+3\right)^{2}}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

Since $\frac{\left(y+5\right)\left(y^{2}+9\right)}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$ and $\frac{6\left(y+3\right)^{2}}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(y+5\right)\left(y^{2}+9\right)+6\left(y+3\right)^{2}}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

Do the multiplications in $\left(y+5\right)\left(y^{2}+9\right)+6\left(y+3\right)^{2}$.

$$\frac{y^{3}+9y+5y^{2}+45+6y^{2}+36y+54}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

Combine like terms in $y^{3}+9y+5y^{2}+45+6y^{2}+36y+54$.

$$\frac{y^{3}+45y+11y^{2}+99}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)^{2}\left(y^{2}+9\right)$ and $y-3$ is $\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)$. Multiply $\frac{y^{3}+45y+11y^{2}+99}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$ times $\frac{y-3}{y-3}$. Multiply $\frac{2}{y-3}$ times $\frac{\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$.

$$\frac{\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Since $\frac{\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$ and $\frac{2\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)-2\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Do the multiplications in $\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)-2\left(y+3\right)^{2}\left(y^{2}+9\right)$.

$$\frac{y^{4}-3y^{3}+45y^{2}-135y+11y^{3}-33y^{2}+99y-297-2y^{4}-18y^{2}-12y^{3}-108y-18y^{2}-162}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Combine like terms in $y^{4}-3y^{3}+45y^{2}-135y+11y^{3}-33y^{2}+99y-297-2y^{4}-18y^{2}-12y^{3}-108y-18y^{2}-162$.

$$\frac{-y^{4}-4y^{3}-24y^{2}-144y-459}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Expand $\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)$.

$$\frac{-y^{4}-4y^{3}-24y^{2}-144y-459}{y^{5}+3y^{4}-81y-243}$$

Expand

$-\frac{y^{4}+4y^{3}+24y^{2}+144y+459}{\left(y-3\right)\left(y^{2}+9\right)\left(y^{2}+6y+9\right)}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{16y}{8y^{2}-24y}$.

$$\frac{y+5}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{16y}{8y\left(y-3\right)}$$

Cancel out $8y$ in both numerator and denominator.

$$\frac{y+5}{y^{2}+6y+9}+\frac{6}{y^{2}+9}-\frac{2}{y-3}$$

Factor $y^{2}+6y+9$.

$$\frac{y+5}{\left(y+3\right)^{2}}+\frac{6}{y^{2}+9}-\frac{2}{y-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)^{2}$ and $y^{2}+9$ is $\left(y+3\right)^{2}\left(y^{2}+9\right)$. Multiply $\frac{y+5}{\left(y+3\right)^{2}}$ times $\frac{y^{2}+9}{y^{2}+9}$. Multiply $\frac{6}{y^{2}+9}$ times $\frac{\left(y+3\right)^{2}}{\left(y+3\right)^{2}}$.

$$\frac{\left(y+5\right)\left(y^{2}+9\right)}{\left(y+3\right)^{2}\left(y^{2}+9\right)}+\frac{6\left(y+3\right)^{2}}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

Since $\frac{\left(y+5\right)\left(y^{2}+9\right)}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$ and $\frac{6\left(y+3\right)^{2}}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(y+5\right)\left(y^{2}+9\right)+6\left(y+3\right)^{2}}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

Do the multiplications in $\left(y+5\right)\left(y^{2}+9\right)+6\left(y+3\right)^{2}$.

$$\frac{y^{3}+9y+5y^{2}+45+6y^{2}+36y+54}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

Combine like terms in $y^{3}+9y+5y^{2}+45+6y^{2}+36y+54$.

$$\frac{y^{3}+45y+11y^{2}+99}{\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2}{y-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)^{2}\left(y^{2}+9\right)$ and $y-3$ is $\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)$. Multiply $\frac{y^{3}+45y+11y^{2}+99}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$ times $\frac{y-3}{y-3}$. Multiply $\frac{2}{y-3}$ times $\frac{\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y+3\right)^{2}\left(y^{2}+9\right)}$.

$$\frac{\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}-\frac{2\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Since $\frac{\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$ and $\frac{2\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)-2\left(y+3\right)^{2}\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Do the multiplications in $\left(y^{3}+45y+11y^{2}+99\right)\left(y-3\right)-2\left(y+3\right)^{2}\left(y^{2}+9\right)$.

$$\frac{y^{4}-3y^{3}+45y^{2}-135y+11y^{3}-33y^{2}+99y-297-2y^{4}-18y^{2}-12y^{3}-108y-18y^{2}-162}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Combine like terms in $y^{4}-3y^{3}+45y^{2}-135y+11y^{3}-33y^{2}+99y-297-2y^{4}-18y^{2}-12y^{3}-108y-18y^{2}-162$.

$$\frac{-y^{4}-4y^{3}-24y^{2}-144y-459}{\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)}$$

Expand $\left(y-3\right)\left(y+3\right)^{2}\left(y^{2}+9\right)$.

$$\frac{-y^{4}-4y^{3}-24y^{2}-144y-459}{y^{5}+3y^{4}-81y-243}$$