Algebra Examples

$f (x) = \frac{x^{2} + x - 6}{x^{2} + x - 20}$

Step 1

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

$y = \frac{{(- 2)}^{2} - 2 - 6}{{(- 2)}^{2} - 2 - 20}$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = \frac{{(- 2)}^{2} - 2 - 6}{{(- 2)}^{2} - 2 - 20}$

Step 2.2

Simplify $\frac{{(- 2)}^{2} - 2 - 6}{{(- 2)}^{2} - 2 - 20}$ .

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

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

$y = \frac{4 - 2 - 6}{{(- 2)}^{2} - 2 - 20}$

Step 2.2.1.2

Subtract $2$ from $4$ .

$y = \frac{2 - 6}{{(- 2)}^{2} - 2 - 20}$

Step 2.2.1.3

Subtract $6$ from $2$ .

$y = \frac{- 4}{{(- 2)}^{2} - 2 - 20}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

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

$y = \frac{- 4}{4 - 2 - 20}$

Step 2.2.2.2

Subtract $2$ from $4$ .

$y = \frac{- 4}{2 - 20}$

Step 2.2.2.3

Subtract $20$ from $2$ .

$y = \frac{- 4}{- 18}$

Step 2.2.3

Cancel the common factor of $- 4$ and $- 18$ .

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Step 2.2.3.1

Factor $- 2$ out of $- 4$ .

$y = \frac{- 2 (2)}{- 18}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $- 2$ out of $- 18$ .

$y = \frac{- 2 \cdot 2}{- 2 \cdot 9}$

Step 2.2.3.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 2}{- 2 \cdot 9}$

Step 2.2.3.2.3

Rewrite the expression.

$y = \frac{2}{9}$

Step 3

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

$y = \frac{{(- 1)}^{2} - 1 - 6}{{(- 1)}^{2} - 1 - 20}$

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

$y = \frac{{(- 1)}^{2} - 1 - 6}{{(- 1)}^{2} - 1 - 20}$

Step 4.2

Simplify $\frac{{(- 1)}^{2} - 1 - 6}{{(- 1)}^{2} - 1 - 20}$ .

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

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

$y = \frac{1 - 1 - 6}{{(- 1)}^{2} - 1 - 20}$

Step 4.2.1.2

Subtract $1$ from $1$ .

$y = \frac{0 - 6}{{(- 1)}^{2} - 1 - 20}$

Step 4.2.1.3

Subtract $6$ from $0$ .

$y = \frac{- 6}{{(- 1)}^{2} - 1 - 20}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

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

$y = \frac{- 6}{1 - 1 - 20}$

Step 4.2.2.2

Subtract $1$ from $1$ .

$y = \frac{- 6}{0 - 20}$

Step 4.2.2.3

Subtract $20$ from $0$ .

$y = \frac{- 6}{- 20}$

Step 4.2.3

Cancel the common factor of $- 6$ and $- 20$ .

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Step 4.2.3.1

Factor $- 2$ out of $- 6$ .

$y = \frac{- 2 (3)}{- 20}$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $- 2$ out of $- 20$ .

$y = \frac{- 2 \cdot 3}{- 2 \cdot 10}$

Step 4.2.3.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 3}{- 2 \cdot 10}$

Step 4.2.3.2.3

Rewrite the expression.

$y = \frac{3}{10}$

Step 5

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

$y = \frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Step 6.2

Remove parentheses.

$y = \frac{0^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Step 6.3

Remove parentheses.

$y = \frac{0^{2} + 0 - 6}{0^{2} + 0 - 20}$

Step 6.4

Remove parentheses.

$y = \frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$

Step 6.5

Simplify $\frac{{(0)}^{2} + 0 - 6}{{(0)}^{2} + 0 - 20}$ .

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

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

$y = \frac{0 + 0 - 6}{0^{2} + 0 - 20}$

Step 6.5.1.2

Add $0$ and $0$ .

$y = \frac{0 - 6}{0^{2} + 0 - 20}$

Step 6.5.1.3

Subtract $6$ from $0$ .

$y = \frac{- 6}{0^{2} + 0 - 20}$

Step 6.5.2

Simplify the denominator.

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Step 6.5.2.1

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

$y = \frac{- 6}{0 + 0 - 20}$

Step 6.5.2.2

Add $0$ and $0$ .

$y = \frac{- 6}{0 - 20}$

Step 6.5.2.3

Subtract $20$ from $0$ .

$y = \frac{- 6}{- 20}$

Step 6.5.3

Cancel the common factor of $- 6$ and $- 20$ .

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Step 6.5.3.1

Factor $- 2$ out of $- 6$ .

$y = \frac{- 2 (3)}{- 20}$

Step 6.5.3.2

Cancel the common factors.

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Step 6.5.3.2.1

Factor $- 2$ out of $- 20$ .

$y = \frac{- 2 \cdot 3}{- 2 \cdot 10}$

Step 6.5.3.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 3}{- 2 \cdot 10}$

Step 6.5.3.2.3

Rewrite the expression.

$y = \frac{3}{10}$

Step 7

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

$y = \frac{{(1)}^{2} + 1 - 6}{{(1)}^{2} + 1 - 20}$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = \frac{{(1)}^{2} + 1 - 6}{{(1)}^{2} + 1 - 20}$

Step 8.2

Remove parentheses.

$y = \frac{1^{2} + 1 - 6}{{(1)}^{2} + 1 - 20}$

Step 8.3

Remove parentheses.

$y = \frac{1^{2} + 1 - 6}{1^{2} + 1 - 20}$

Step 8.4

Remove parentheses.

$y = \frac{{(1)}^{2} + 1 - 6}{{(1)}^{2} + 1 - 20}$

Step 8.5

Simplify $\frac{{(1)}^{2} + 1 - 6}{{(1)}^{2} + 1 - 20}$ .

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

One to any power is one.

$y = \frac{1 + 1 - 6}{1^{2} + 1 - 20}$

Step 8.5.1.2

Add $1$ and $1$ .

$y = \frac{2 - 6}{1^{2} + 1 - 20}$

Step 8.5.1.3

Subtract $6$ from $2$ .

$y = \frac{- 4}{1^{2} + 1 - 20}$

Step 8.5.2

Simplify the denominator.

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Step 8.5.2.1

One to any power is one.

$y = \frac{- 4}{1 + 1 - 20}$

Step 8.5.2.2

Add $1$ and $1$ .

$y = \frac{- 4}{2 - 20}$

Step 8.5.2.3

Subtract $20$ from $2$ .

$y = \frac{- 4}{- 18}$

Step 8.5.3

Cancel the common factor of $- 4$ and $- 18$ .

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Step 8.5.3.1

Factor $- 2$ out of $- 4$ .

$y = \frac{- 2 (2)}{- 18}$

Step 8.5.3.2

Cancel the common factors.

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Step 8.5.3.2.1

Factor $- 2$ out of $- 18$ .

$y = \frac{- 2 \cdot 2}{- 2 \cdot 9}$

Step 8.5.3.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 2}{- 2 \cdot 9}$

Step 8.5.3.2.3

Rewrite the expression.

$y = \frac{2}{9}$

Step 9

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

$y = \frac{{(2)}^{2} + 2 - 6}{{(2)}^{2} + 2 - 20}$

Step 10

Solve the equation for $y$ .

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Step 10.1

Remove parentheses.

$y = \frac{{(2)}^{2} + 2 - 6}{{(2)}^{2} + 2 - 20}$

Step 10.2

Remove parentheses.

$y = \frac{2^{2} + 2 - 6}{{(2)}^{2} + 2 - 20}$

Step 10.3

Remove parentheses.

$y = \frac{2^{2} + 2 - 6}{2^{2} + 2 - 20}$

Step 10.4

Remove parentheses.

$y = \frac{{(2)}^{2} + 2 - 6}{{(2)}^{2} + 2 - 20}$

Step 10.5

Simplify $\frac{{(2)}^{2} + 2 - 6}{{(2)}^{2} + 2 - 20}$ .

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Step 10.5.1

Cancel the common factor of ${(2)}^{2} + 2 - 6$ and ${(2)}^{2} + 2 - 20$ .

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Step 10.5.1.1

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

$y = \frac{2 \cdot 2 + 2 - 6}{{(2)}^{2} + 2 - 20}$

Step 10.5.1.2

Factor $2$ out of $2$ .

$y = \frac{2 \cdot 2 + 2 \cdot 1 - 6}{2^{2} + 2 - 20}$

Step 10.5.1.3

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

$y = \frac{2 \cdot (2 + 1) - 6}{{(2)}^{2} + 2 - 20}$

Step 10.5.1.4

Factor $2$ out of $- 6$ .

$y = \frac{2 \cdot (2 + 1) + 2 \cdot - 3}{2^{2} + 2 - 20}$

Step 10.5.1.5

Factor $2$ out of $2 \cdot (2 + 1) + 2 (- 3)$ .

$y = \frac{2 \cdot (2 + 1 - 3)}{{(2)}^{2} + 2 - 20}$

Step 10.5.1.6

Cancel the common factors.

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Step 10.5.1.6.1

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

$y = \frac{2 \cdot (2 + 1 - 3)}{2 \cdot 2 + 2 - 20}$

Step 10.5.1.6.2

Factor $2$ out of $2$ .

$y = \frac{2 \cdot (2 + 1 - 3)}{2 \cdot 2 + 2 (1) - 20}$

Step 10.5.1.6.3

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

$y = \frac{2 \cdot (2 + 1 - 3)}{2 \cdot (2 + 1) - 20}$

Step 10.5.1.6.4

Factor $2$ out of $- 20$ .

$y = \frac{2 \cdot (2 + 1 - 3)}{2 \cdot (2 + 1) + 2 (- 10)}$

Step 10.5.1.6.5

Factor $2$ out of $2 \cdot (2 + 1) + 2 (- 10)$ .

$y = \frac{2 \cdot (2 + 1 - 3)}{2 \cdot (2 + 1 - 10)}$

Step 10.5.1.6.6

Cancel the common factor.

$y = \frac{2 \cdot (2 + 1 - 3)}{2 \cdot (2 + 1 - 10)}$

Step 10.5.1.6.7

Rewrite the expression.

$y = \frac{2 + 1 - 3}{2 + 1 - 10}$

Step 10.5.2

Simplify the numerator.

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Step 10.5.2.1

Add $2$ and $1$ .

$y = \frac{3 - 3}{2 + 1 - 10}$

Step 10.5.2.2

Subtract $3$ from $3$ .

$y = \frac{0}{2 + 1 - 10}$

Step 10.5.3

Simplify the denominator.

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Step 10.5.3.1

Add $2$ and $1$ .

$y = \frac{0}{3 - 10}$

Step 10.5.3.2

Subtract $10$ from $3$ .

$y = \frac{0}{- 7}$

Step 10.5.4

Divide $0$ by $- 7$ .

$y = 0$

Step 11

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

$\begin{array}{c} x & y \\ - 2 & \frac{2}{9} \\ - 1 & \frac{3}{10} \\ 0 & \frac{3}{10} \\ 1 & \frac{2}{9} \\ 2 & 0 \end{array}$

Step 12