Algebra Examples

$\frac{x + 3}{- 3 x^{2} - 9 x}$

Step 1

Factor $3 x$ out of $- 3 x^{2} - 9 x$ .

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Step 1.1

Factor $3 x$ out of $- 3 x^{2}$ .

$\frac{x + 3}{3 x (- x) - 9 x}$

Step 1.2

Factor $3 x$ out of $- 9 x$ .

$\frac{x + 3}{3 x (- x) + 3 x (- 3)}$

Step 1.3

Factor $3 x$ out of $3 x (- x) + 3 x (- 3)$ .

$\frac{x + 3}{3 x (- x - 3)}$

Step 2

Cancel the common factor of $x + 3$ and $- x - 3$ .

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Step 2.1

Factor $- 1$ out of $x$ .

$\frac{- 1 (- x) + 3}{3 x (- x - 3)}$

Step 2.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 1 (- x) - 1 (- 3)}{3 x (- x - 3)}$

Step 2.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 3)$ .

$\frac{- 1 (- x - 3)}{3 x (- x - 3)}$

Step 2.4

Cancel the common factor.

$\frac{- 1 (- x - 3)}{3 x (- x - 3)}$

Step 2.5

Rewrite the expression.

$\frac{- 1}{3 x}$

Step 3

Move the negative in front of the fraction.

$- \frac{1}{3 x}$

Step 4

To find the holes in the graph, look at the denominator factors that were cancelled.

$- x - 3$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $- \frac{1}{3 x}$ .

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Step 5.1

Set $- x - 3$ equal to $0$ .

$- x - 3 = 0$

Step 5.2

Solve $- x - 3 = 0$ for $x$ .

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Step 5.2.1

Add $3$ to both sides of the equation.

$- x = 3$

Step 5.2.2

Divide each term in $- x = 3$ by $- 1$ and simplify.

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Step 5.2.2.1

Divide each term in $- x = 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{3}{- 1}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{3}{- 1}$

Step 5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{3}{- 1}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide $3$ by $- 1$ .

$x = - 3$

Step 5.3

Substitute $- 3$ for $x$ in $- \frac{1}{3 x}$ and simplify.

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Step 5.3.1

Substitute $- 3$ for $x$ to find the $y$ coordinate of the hole.

$- \frac{1}{3 \cdot - 3}$

Step 5.3.2

Simplify.

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Step 5.3.2.1

Multiply $3$ by $- 3$ .

$- \frac{1}{- 9}$

Step 5.3.2.2

Move the negative in front of the fraction.

$- - \frac{1}{9}$

Step 5.3.2.3

Multiply $- - \frac{1}{9}$ .

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Step 5.3.2.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{9})$

Step 5.3.2.3.2

Multiply $\frac{1}{9}$ by $1$ .

$\frac{1}{9}$

Step 5.4

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 3, \frac{1}{9})$

Step 6