Algebra Examples

$f (x) = \frac{- 5 x - 15}{x^{2} + 2 x - 15}$

Step 1

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

$y = \frac{- 5 \cdot - 2 - 15}{{(- 2)}^{2} + 2 (- 2) - 15}$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = \frac{- 5 \cdot - 2 - 15}{{(- 2)}^{2} + 2 (- 2) - 15}$

Step 2.2

Simplify $\frac{- 5 \cdot - 2 - 15}{{(- 2)}^{2} + 2 (- 2) - 15}$ .

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $- 5$ by $- 2$ .

$y = \frac{10 - 15}{{(- 2)}^{2} + 2 (- 2) - 15}$

Step 2.2.1.2

Subtract $15$ from $10$ .

$y = \frac{- 5}{{(- 2)}^{2} + 2 (- 2) - 15}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

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

$y = \frac{- 5}{4 + 2 (- 2) - 15}$

Step 2.2.2.2

Multiply $2$ by $- 2$ .

$y = \frac{- 5}{4 - 4 - 15}$

Step 2.2.2.3

Subtract $4$ from $4$ .

$y = \frac{- 5}{0 - 15}$

Step 2.2.2.4

Subtract $15$ from $0$ .

$y = \frac{- 5}{- 15}$

Step 2.2.3

Cancel the common factor of $- 5$ and $- 15$ .

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Step 2.2.3.1

Factor $- 5$ out of $- 5$ .

$y = \frac{- 5 (1)}{- 15}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $- 5$ out of $- 15$ .

$y = \frac{- 5 \cdot 1}{- 5 \cdot 3}$

Step 2.2.3.2.2

Cancel the common factor.

$y = \frac{- 5 \cdot 1}{- 5 \cdot 3}$

Step 2.2.3.2.3

Rewrite the expression.

$y = \frac{1}{3}$

Step 3

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

$y = \frac{- 5 \cdot - 1 - 15}{{(- 1)}^{2} + 2 (- 1) - 15}$

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

$y = \frac{- 5 \cdot - 1 - 15}{{(- 1)}^{2} + 2 (- 1) - 15}$

Step 4.2

Simplify $\frac{- 5 \cdot - 1 - 15}{{(- 1)}^{2} + 2 (- 1) - 15}$ .

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Multiply $- 5$ by $- 1$ .

$y = \frac{5 - 15}{{(- 1)}^{2} + 2 (- 1) - 15}$

Step 4.2.1.2

Subtract $15$ from $5$ .

$y = \frac{- 10}{{(- 1)}^{2} + 2 (- 1) - 15}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

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

$y = \frac{- 10}{1 + 2 (- 1) - 15}$

Step 4.2.2.2

Multiply $2$ by $- 1$ .

$y = \frac{- 10}{1 - 2 - 15}$

Step 4.2.2.3

Subtract $2$ from $1$ .

$y = \frac{- 10}{- 1 - 15}$

Step 4.2.2.4

Subtract $15$ from $- 1$ .

$y = \frac{- 10}{- 16}$

Step 4.2.3

Cancel the common factor of $- 10$ and $- 16$ .

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Step 4.2.3.1

Factor $- 2$ out of $- 10$ .

$y = \frac{- 2 (5)}{- 16}$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $- 2$ out of $- 16$ .

$y = \frac{- 2 \cdot 5}{- 2 \cdot 8}$

Step 4.2.3.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 5}{- 2 \cdot 8}$

Step 4.2.3.2.3

Rewrite the expression.

$y = \frac{5}{8}$

Step 5

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

$y = \frac{- 5 \cdot 0 - 15}{{(0)}^{2} + 2 (0) - 15}$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \frac{- 5 \cdot 0 - 15}{0^{2} + 2 (0) - 15}$

Step 6.2

Remove parentheses.

$y = \frac{- 5 \cdot 0 - 15}{{(0)}^{2} + 2 (0) - 15}$

Step 6.3

Simplify $\frac{- 5 \cdot 0 - 15}{{(0)}^{2} + 2 (0) - 15}$ .

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Step 6.3.1

Simplify the numerator.

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Step 6.3.1.1

Multiply $- 5$ by $0$ .

$y = \frac{0 - 15}{0^{2} + 2 (0) - 15}$

Step 6.3.1.2

Subtract $15$ from $0$ .

$y = \frac{- 15}{0^{2} + 2 (0) - 15}$

Step 6.3.2

Simplify the denominator.

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Step 6.3.2.1

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

$y = \frac{- 15}{0 + 2 (0) - 15}$

Step 6.3.2.2

Multiply $2$ by $0$ .

$y = \frac{- 15}{0 + 0 - 15}$

Step 6.3.2.3

Add $0$ and $0$ .

$y = \frac{- 15}{0 - 15}$

Step 6.3.2.4

Subtract $15$ from $0$ .

$y = \frac{- 15}{- 15}$

Step 6.3.3

Divide $- 15$ by $- 15$ .

$y = 1$

Step 7

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

$y = \frac{- 5 \cdot 1 - 15}{{(1)}^{2} + 2 (1) - 15}$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = \frac{- 5 \cdot 1 - 15}{1^{2} + 2 (1) - 15}$

Step 8.2

Remove parentheses.

$y = \frac{- 5 \cdot 1 - 15}{{(1)}^{2} + 2 (1) - 15}$

Step 8.3

Simplify $\frac{- 5 \cdot 1 - 15}{{(1)}^{2} + 2 (1) - 15}$ .

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Multiply $- 5$ by $1$ .

$y = \frac{- 5 - 15}{1^{2} + 2 (1) - 15}$

Step 8.3.1.2

Subtract $15$ from $- 5$ .

$y = \frac{- 20}{1^{2} + 2 (1) - 15}$

Step 8.3.2

Simplify the denominator.

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Step 8.3.2.1

One to any power is one.

$y = \frac{- 20}{1 + 2 (1) - 15}$

Step 8.3.2.2

Multiply $2$ by $1$ .

$y = \frac{- 20}{1 + 2 - 15}$

Step 8.3.2.3

Add $1$ and $2$ .

$y = \frac{- 20}{3 - 15}$

Step 8.3.2.4

Subtract $15$ from $3$ .

$y = \frac{- 20}{- 12}$

Step 8.3.3

Cancel the common factor of $- 20$ and $- 12$ .

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Step 8.3.3.1

Factor $- 4$ out of $- 20$ .

$y = \frac{- 4 (5)}{- 12}$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $- 4$ out of $- 12$ .

$y = \frac{- 4 \cdot 5}{- 4 \cdot 3}$

Step 8.3.3.2.2

Cancel the common factor.

$y = \frac{- 4 \cdot 5}{- 4 \cdot 3}$

Step 8.3.3.2.3

Rewrite the expression.

$y = \frac{5}{3}$

Step 9

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

$y = \frac{- 5 \cdot 2 - 15}{{(2)}^{2} + 2 (2) - 15}$

Step 10

Solve the equation for $y$ .

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Step 10.1

Remove parentheses.

$y = \frac{- 5 \cdot 2 - 15}{2^{2} + 2 (2) - 15}$

Step 10.2

Remove parentheses.

$y = \frac{- 5 \cdot 2 - 15}{{(2)}^{2} + 2 (2) - 15}$

Step 10.3

Simplify $\frac{- 5 \cdot 2 - 15}{{(2)}^{2} + 2 (2) - 15}$ .

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Step 10.3.1

Simplify the numerator.

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Step 10.3.1.1

Multiply $- 5$ by $2$ .

$y = \frac{- 10 - 15}{2^{2} + 2 (2) - 15}$

Step 10.3.1.2

Subtract $15$ from $- 10$ .

$y = \frac{- 25}{2^{2} + 2 (2) - 15}$

Step 10.3.2

Simplify the denominator.

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Step 10.3.2.1

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

$y = \frac{- 25}{4 + 2 (2) - 15}$

Step 10.3.2.2

Multiply $2$ by $2$ .

$y = \frac{- 25}{4 + 4 - 15}$

Step 10.3.2.3

Add $4$ and $4$ .

$y = \frac{- 25}{8 - 15}$

Step 10.3.2.4

Subtract $15$ from $8$ .

$y = \frac{- 25}{- 7}$

Step 10.3.3

Dividing two negative values results in a positive value.

$y = \frac{25}{7}$

Step 11

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

$\begin{array}{c} x & y \\ - 2 & \frac{1}{3} \\ - 1 & \frac{5}{8} \\ 0 & 1 \\ 1 & \frac{5}{3} \\ 2 & \frac{25}{7} \end{array}$

Step 12