Algebra Examples

$\frac{(x + 2) (x - 4)}{(x + 4) (x - 5) (x + 1)}$

Step 1

Write $\frac{(x + 2) (x - 4)}{(x + 4) (x - 5) (x + 1)}$ as an equation.

$y = \frac{(x + 2) (x - 4)}{(x + 4) (x - 5) (x + 1)}$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{(x + 2) (x - 4)}{(x + 4) (x - 5) (x + 1)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$(x + 2) (x - 4) = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x - 4 = 0$

Step 2.2.2.2

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.2.2.2.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.2.2.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.2.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.2.2.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.2.2.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.2.2.4

The final solution is all the values that make $(x + 2) (x - 4) = 0$ true.

$x = - 2, 4$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (4, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{((0) + 2) ((0) - 4)}{((0) + 4) ((0) - 5) ((0) + 1)}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{(0 + 2) ((0) - 4)}{((0) + 4) ((0) - 5) ((0) + 1)}$

Step 3.2.2

Remove parentheses.

$y = \frac{(0 + 2) (0 - 4)}{((0) + 4) ((0) - 5) ((0) + 1)}$

Step 3.2.3

Remove parentheses.

$y = \frac{(0 + 2) (0 - 4)}{(0 + 4) ((0) - 5) ((0) + 1)}$

Step 3.2.4

Remove parentheses.

$y = \frac{(0 + 2) (0 - 4)}{(0 + 4) (0 - 5) ((0) + 1)}$

Step 3.2.5

Remove parentheses.

$y = \frac{(0 + 2) (0 - 4)}{(0 + 4) (0 - 5) (0 + 1)}$

Step 3.2.6

Remove parentheses.

$y = \frac{((0) + 2) ((0) - 4)}{((0) + 4) ((0) - 5) ((0) + 1)}$

Step 3.2.7

Simplify $\frac{((0) + 2) ((0) - 4)}{((0) + 4) ((0) - 5) ((0) + 1)}$ .

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Step 3.2.7.1

Reduce the expression by cancelling the common factors.

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Step 3.2.7.1.1

Cancel the common factor of $(0) + 2$ and $(0) + 4$ .

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Step 3.2.7.1.1.1

Factor $2$ out of $((0) + 2) ((0) - 4)$ .

$y = \frac{2 ((0 + 1) ((0) - 4))}{((0) + 4) ((0) - 5) ((0) + 1)}$

Step 3.2.7.1.1.2

Cancel the common factors.

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Step 3.2.7.1.1.2.1

Factor $2$ out of $((0) + 4) ((0) - 5) ((0) + 1)$ .

$y = \frac{2 ((0 + 1) ((0) - 4))}{2 (((0 + 2) ((0) - 5)) ((0) + 1))}$

Step 3.2.7.1.1.2.2

Cancel the common factor.

$y = \frac{2 ((0 + 1) ((0) - 4))}{2 (((0 + 2) ((0) - 5)) ((0) + 1))}$

Step 3.2.7.1.1.2.3

Rewrite the expression.

$y = \frac{(0 + 1) ((0) - 4)}{((0 + 2) ((0) - 5)) ((0) + 1)}$

Step 3.2.7.1.2

Cancel the common factor of $0 + 1$ and $(0) + 1$ .

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Step 3.2.7.1.2.1

Cancel the common factor.

$y = \frac{(0 + 1) ((0) - 4)}{(0 + 2) ((0) - 5) ((0) + 1)}$

Step 3.2.7.1.2.2

Rewrite the expression.

$y = \frac{(0) - 4}{(0 + 2) ((0) - 5)}$

Step 3.2.7.1.3

Cancel the common factor of $(0) - 4$ and $0 + 2$ .

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Step 3.2.7.1.3.1

Factor $2$ out of $0$ .

$y = \frac{2 (0) - 4}{(0 + 2) ((0) - 5)}$

Step 3.2.7.1.3.2

Factor $2$ out of $- 4$ .

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

Step 3.2.7.1.3.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot - 2$ .

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

Step 3.2.7.1.3.4

Cancel the common factors.

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Step 3.2.7.1.3.4.1

Factor $2$ out of $(0 + 2) ((0) - 5)$ .

$y = \frac{2 \cdot (0 - 2)}{2 ((0 + 1) ((0) - 5))}$

Step 3.2.7.1.3.4.2

Cancel the common factor.

$y = \frac{2 \cdot (0 - 2)}{2 ((0 + 1) ((0) - 5))}$

Step 3.2.7.1.3.4.3

Rewrite the expression.

$y = \frac{0 - 2}{(0 + 1) ((0) - 5)}$

Step 3.2.7.1.4

Subtract $2$ from $0$ .

$y = \frac{- 2}{(0 + 1) (0 - 5)}$

Step 3.2.7.2

Simplify the denominator.

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Step 3.2.7.2.1

Add $0$ and $1$ .

$y = \frac{- 2}{1 (0 - 5)}$

Step 3.2.7.2.2

Multiply $0 - 5$ by $1$ .

$y = \frac{- 2}{0 - 5}$

Step 3.2.7.2.3

Subtract $5$ from $0$ .

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

Step 3.2.7.3

Dividing two negative values results in a positive value.

$y = \frac{2}{5}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{2}{5})$

Step 4

List the intersections.

x-intercept(s): $(- 2, 0), (4, 0)$

y-intercept(s): $(0, \frac{2}{5})$

Step 5