Algebra Examples

$f (x) = \frac{x^{2} + 6 x + 9}{x^{3} + 7 x^{2}}$

Step 1

Substitute $- 2$ for $x$ and find the result for $y$ .

$y = \frac{{(- 2)}^{2} + 6 (- 2) + 9}{{(- 2)}^{3} + 7 {(- 2)}^{2}}$

Step 2

Simplify $\frac{{(- 2)}^{2} + 6 (- 2) + 9}{{(- 2)}^{3} + 7 {(- 2)}^{2}}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Raise $- 2$ to the power of $2$ .

$y = \frac{4 + 6 (- 2) + 9}{{(- 2)}^{3} + 7 {(- 2)}^{2}}$

Step 2.1.2

Multiply $6$ by $- 2$ .

$y = \frac{4 - 12 + 9}{{(- 2)}^{3} + 7 {(- 2)}^{2}}$

Step 2.1.3

Subtract $12$ from $4$ .

$y = \frac{- 8 + 9}{{(- 2)}^{3} + 7 {(- 2)}^{2}}$

Step 2.1.4

Add $- 8$ and $9$ .

$y = \frac{1}{{(- 2)}^{3} + 7 {(- 2)}^{2}}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Raise $- 2$ to the power of $3$ .

$y = \frac{1}{- 8 + 7 {(- 2)}^{2}}$

Step 2.2.2

Raise $- 2$ to the power of $2$ .

$y = \frac{1}{- 8 + 7 \cdot 4}$

Step 2.2.3

Multiply $7$ by $4$ .

$y = \frac{1}{- 8 + 28}$

Step 2.2.4

Add $- 8$ and $28$ .

$y = \frac{1}{20}$

Step 3

Substitute $- 1$ for $x$ and find the result for $y$ .

$y = \frac{{(- 1)}^{2} + 6 (- 1) + 9}{{(- 1)}^{3} + 7 {(- 1)}^{2}}$

Step 4

Simplify $\frac{{(- 1)}^{2} + 6 (- 1) + 9}{{(- 1)}^{3} + 7 {(- 1)}^{2}}$ .

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 1$ to the power of $2$ .

$y = \frac{1 + 6 (- 1) + 9}{{(- 1)}^{3} + 7 {(- 1)}^{2}}$

Step 4.1.2

Multiply $6$ by $- 1$ .

$y = \frac{1 - 6 + 9}{{(- 1)}^{3} + 7 {(- 1)}^{2}}$

Step 4.1.3

Subtract $6$ from $1$ .

$y = \frac{- 5 + 9}{{(- 1)}^{3} + 7 {(- 1)}^{2}}$

Step 4.1.4

Add $- 5$ and $9$ .

$y = \frac{4}{{(- 1)}^{3} + 7 {(- 1)}^{2}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Raise $- 1$ to the power of $3$ .

$y = \frac{4}{- 1 + 7 {(- 1)}^{2}}$

Step 4.2.2

Raise $- 1$ to the power of $2$ .

$y = \frac{4}{- 1 + 7 \cdot 1}$

Step 4.2.3

Multiply $7$ by $1$ .

$y = \frac{4}{- 1 + 7}$

Step 4.2.4

Add $- 1$ and $7$ .

$y = \frac{4}{6}$

Step 4.3

Cancel the common factor of $4$ and $6$ .

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Step 4.3.1

Factor $2$ out of $4$ .

$y = \frac{2 (2)}{6}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot 2}{2 \cdot 3}$

Step 4.3.2.2

Cancel the common factor.

$y = \frac{2 \cdot 2}{2 \cdot 3}$

Step 4.3.2.3

Rewrite the expression.

$y = \frac{2}{3}$

Step 5

Substitute $1$ for $x$ and find the result for $y$ .

$y = \frac{{(1)}^{2} + 6 (1) + 9}{{(1)}^{3} + 7 {(1)}^{2}}$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \frac{1^{2} + 6 (1) + 9}{{(1)}^{3} + 7 {(1)}^{2}}$

Step 6.2

Remove parentheses.

$y = \frac{1^{2} + 6 (1) + 9}{1^{3} + 7 {(1)}^{2}}$

Step 6.3

Remove parentheses.

$y = \frac{1^{2} + 6 (1) + 9}{1^{3} + 7 \cdot 1^{2}}$

Step 6.4

Remove parentheses.

$y = \frac{{(1)}^{2} + 6 (1) + 9}{{(1)}^{3} + 7 {(1)}^{2}}$

Step 6.5

Simplify $\frac{{(1)}^{2} + 6 (1) + 9}{{(1)}^{3} + 7 {(1)}^{2}}$ .

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

One to any power is one.

$y = \frac{1 + 6 (1) + 9}{1^{3} + 7 \cdot 1^{2}}$

Step 6.5.1.2

Multiply $6$ by $1$ .

$y = \frac{1 + 6 + 9}{1^{3} + 7 \cdot 1^{2}}$

Step 6.5.1.3

Add $1$ and $6$ .

$y = \frac{7 + 9}{1^{3} + 7 \cdot 1^{2}}$

Step 6.5.1.4

Add $7$ and $9$ .

$y = \frac{16}{1^{3} + 7 \cdot 1^{2}}$

Step 6.5.2

Simplify the denominator.

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Step 6.5.2.1

One to any power is one.

$y = \frac{16}{1 + 7 \cdot 1^{2}}$

Step 6.5.2.2

One to any power is one.

$y = \frac{16}{1 + 7 \cdot 1}$

Step 6.5.2.3

Multiply $7$ by $1$ .

$y = \frac{16}{1 + 7}$

Step 6.5.2.4

Add $1$ and $7$ .

$y = \frac{16}{8}$

Step 6.5.3

Divide $16$ by $8$ .

$y = 2$

Step 7

Substitute $2$ for $x$ and find the result for $y$ .

$y = \frac{{(2)}^{2} + 6 (2) + 9}{{(2)}^{3} + 7 {(2)}^{2}}$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = \frac{2^{2} + 6 (2) + 9}{{(2)}^{3} + 7 {(2)}^{2}}$

Step 8.2

Remove parentheses.

$y = \frac{2^{2} + 6 (2) + 9}{2^{3} + 7 {(2)}^{2}}$

Step 8.3

Remove parentheses.

$y = \frac{2^{2} + 6 (2) + 9}{2^{3} + 7 \cdot 2^{2}}$

Step 8.4

Remove parentheses.

$y = \frac{{(2)}^{2} + 6 (2) + 9}{{(2)}^{3} + 7 {(2)}^{2}}$

Step 8.5

Simplify $\frac{{(2)}^{2} + 6 (2) + 9}{{(2)}^{3} + 7 {(2)}^{2}}$ .

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

Raise $2$ to the power of $2$ .

$y = \frac{4 + 6 (2) + 9}{2^{3} + 7 \cdot 2^{2}}$

Step 8.5.1.2

Multiply $6$ by $2$ .

$y = \frac{4 + 12 + 9}{2^{3} + 7 \cdot 2^{2}}$

Step 8.5.1.3

Add $4$ and $12$ .

$y = \frac{16 + 9}{2^{3} + 7 \cdot 2^{2}}$

Step 8.5.1.4

Add $16$ and $9$ .

$y = \frac{25}{2^{3} + 7 \cdot 2^{2}}$

Step 8.5.2

Simplify the denominator.

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Step 8.5.2.1

Raise $2$ to the power of $3$ .

$y = \frac{25}{8 + 7 \cdot 2^{2}}$

Step 8.5.2.2

Raise $2$ to the power of $2$ .

$y = \frac{25}{8 + 7 \cdot 4}$

Step 8.5.2.3

Multiply $7$ by $4$ .

$y = \frac{25}{8 + 28}$

Step 8.5.2.4

Add $8$ and $28$ .

$y = \frac{25}{36}$

Step 9

This is a table of possible values to use when graphing the equation.

$\begin{array}{c} x & y \\ - 2 & \frac{1}{20} \\ - 1 & \frac{2}{3} \\ 1 & 2 \\ 2 & \frac{25}{36} \end{array}$

Step 10