Algebra Examples

$f (x) = \frac{- 16}{x^{2} - 2 x - 8}$

Step 1

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

$y = \frac{- 16}{{(- 1)}^{2} - 2 \cdot - 1 - 8}$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = \frac{- 16}{{(- 1)}^{2} - 2 \cdot - 1 - 8}$

Step 2.2

Simplify $\frac{- 16}{{(- 1)}^{2} - 2 \cdot - 1 - 8}$ .

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Step 2.2.1

Simplify the denominator.

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Step 2.2.1.1

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

$y = \frac{- 16}{1 - 2 \cdot - 1 - 8}$

Step 2.2.1.2

Multiply $- 2$ by $- 1$ .

$y = \frac{- 16}{1 + 2 - 8}$

Step 2.2.1.3

Add $1$ and $2$ .

$y = \frac{- 16}{3 - 8}$

Step 2.2.1.4

Subtract $8$ from $3$ .

$y = \frac{- 16}{- 5}$

Step 2.2.2

Dividing two negative values results in a positive value.

$y = \frac{16}{5}$

Step 3

Substitute $0$ for $x$ and find the result for $y$ .

$y = \frac{- 16}{{(0)}^{2} - 2 \cdot 0 - 8}$

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

$y = \frac{- 16}{0^{2} - 2 \cdot 0 - 8}$

Step 4.2

Remove parentheses.

$y = \frac{- 16}{{(0)}^{2} - 2 \cdot 0 - 8}$

Step 4.3

Simplify $\frac{- 16}{{(0)}^{2} - 2 \cdot 0 - 8}$ .

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Step 4.3.1

Simplify the denominator.

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Step 4.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{- 16}{0 - 2 \cdot 0 - 8}$

Step 4.3.1.2

Multiply $- 2$ by $0$ .

$y = \frac{- 16}{0 + 0 - 8}$

Step 4.3.1.3

Add $0$ and $0$ .

$y = \frac{- 16}{0 - 8}$

Step 4.3.1.4

Subtract $8$ from $0$ .

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

Step 4.3.2

Divide $- 16$ by $- 8$ .

$y = 2$

Step 5

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

$y = \frac{- 16}{{(1)}^{2} - 2 \cdot 1 - 8}$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \frac{- 16}{1^{2} - 2 \cdot 1 - 8}$

Step 6.2

Remove parentheses.

$y = \frac{- 16}{{(1)}^{2} - 2 \cdot 1 - 8}$

Step 6.3

Simplify $\frac{- 16}{{(1)}^{2} - 2 \cdot 1 - 8}$ .

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Step 6.3.1

Simplify the denominator.

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Step 6.3.1.1

One to any power is one.

$y = \frac{- 16}{1 - 2 \cdot 1 - 8}$

Step 6.3.1.2

Multiply $- 2$ by $1$ .

$y = \frac{- 16}{1 - 2 - 8}$

Step 6.3.1.3

Subtract $2$ from $1$ .

$y = \frac{- 16}{- 1 - 8}$

Step 6.3.1.4

Subtract $8$ from $- 1$ .

$y = \frac{- 16}{- 9}$

Step 6.3.2

Dividing two negative values results in a positive value.

$y = \frac{16}{9}$

Step 7

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

$y = \frac{- 16}{{(2)}^{2} - 2 \cdot 2 - 8}$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = \frac{- 16}{2^{2} - 2 \cdot 2 - 8}$

Step 8.2

Remove parentheses.

$y = \frac{- 16}{{(2)}^{2} - 2 \cdot 2 - 8}$

Step 8.3

Simplify $\frac{- 16}{{(2)}^{2} - 2 \cdot 2 - 8}$ .

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Step 8.3.1

Reduce the expression by cancelling the common factors.

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Step 8.3.1.1

Cancel the common factor of $- 16$ and ${(2)}^{2} - 2 \cdot 2 - 8$ .

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Step 8.3.1.1.1

Factor $2$ out of $- 16$ .

$y = \frac{2 \cdot - 8}{2^{2} - 2 \cdot 2 - 8}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

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

$y = \frac{2 \cdot - 8}{2 \cdot 2 - 2 \cdot 2 - 8}$

Step 8.3.1.1.2.2

Factor $2$ out of $- 2 \cdot 2$ .

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

Step 8.3.1.1.2.3

Factor $2$ out of $2 \cdot 2 + 2 (- 1 \cdot 2)$ .

$y = \frac{2 \cdot - 8}{2 \cdot (2 - 1 \cdot 2) - 8}$

Step 8.3.1.1.2.4

Factor $2$ out of $- 8$ .

$y = \frac{2 \cdot - 8}{2 \cdot (2 - 1 \cdot 2) + 2 (- 4)}$

Step 8.3.1.1.2.5

Factor $2$ out of $2 \cdot (2 - 1 \cdot 2) + 2 (- 4)$ .

$y = \frac{2 \cdot - 8}{2 \cdot (2 - 1 \cdot 2 - 4)}$

Step 8.3.1.1.2.6

Cancel the common factor.

$y = \frac{2 \cdot - 8}{2 \cdot (2 - 1 \cdot 2 - 4)}$

Step 8.3.1.1.2.7

Rewrite the expression.

$y = \frac{- 8}{2 - 1 \cdot 2 - 4}$

Step 8.3.1.2

Cancel the common factor of $- 8$ and $2 - 1 \cdot 2 - 4$ .

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Step 8.3.1.2.1

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- 8}{- 1 (- 2) - 1 \cdot 2 - 4}$

Step 8.3.1.2.2

Factor $- 1$ out of $- 1 \cdot 2$ .

$y = \frac{- 8}{- 1 (- 2) - 1 (2) - 4}$

Step 8.3.1.2.3

Factor $- 1$ out of $- 1 (- 2) - 1 (2)$ .

$y = \frac{- 8}{- 1 (- 2 + 2) - 4}$

Step 8.3.1.2.4

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

$y = \frac{- 8}{- 1 (- 2 + 2) - 1 \cdot 4}$

Step 8.3.1.2.5

Factor $- 1$ out of $- 1 (- 2 + 2) - 1 \cdot 4$ .

$y = \frac{- 8}{- 1 (- 2 + 2 + 4)}$

Step 8.3.1.2.6

Factor $2$ out of $- 8$ .

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

Step 8.3.1.2.7

Cancel the common factors.

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Step 8.3.1.2.7.1

Factor $2$ out of $- 1 (- 2 + 2 + 4)$ .

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

Step 8.3.1.2.7.2

Cancel the common factor.

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

Step 8.3.1.2.7.3

Rewrite the expression.

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

Step 8.3.2

Simplify the denominator.

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Step 8.3.2.1

Add $- 1$ and $1$ .

$y = \frac{- 4}{- 1 (0 + 2)}$

Step 8.3.2.2

Add $0$ and $2$ .

$y = \frac{- 4}{- 1 \cdot 2}$

Step 8.3.3

Simplify the expression.

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Step 8.3.3.1

Multiply $- 1$ by $2$ .

$y = \frac{- 4}{- 2}$

Step 8.3.3.2

Divide $- 4$ by $- 2$ .

$y = 2$

Step 9

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

$\begin{array}{c} x & y \\ - 1 & \frac{16}{5} \\ 0 & 2 \\ 1 & \frac{16}{9} \\ 2 & 2 \end{array}$

Step 10