Algebra Examples

$f (x) = \frac{(x + 3) (6 - x)}{(x - 2) (x + 3)}$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \frac{(x + 3) (6 - x)}{(x - 2) (x + 3)}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$(x + 3) (6 - x) = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

$x + 3 = 0$

$6 - x = 0$

Step 1.2.2.2

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 1.2.2.2.1

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

$x + 3 = 0$

Step 1.2.2.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.2.3

Set $6 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $6 - x$ equal to $0$ .

$6 - x = 0$

Step 1.2.2.3.2

Solve $6 - x = 0$ for $x$ .

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Step 1.2.2.3.2.1

Subtract $6$ from both sides of the equation.

$- x = - 6$

Step 1.2.2.3.2.2

Divide each term in $- x = - 6$ by $- 1$ and simplify.

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Step 1.2.2.3.2.2.1

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Step 1.2.2.3.2.2.2

Simplify the left side.

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Step 1.2.2.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Step 1.2.2.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Step 1.2.2.3.2.2.3

Simplify the right side.

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Step 1.2.2.3.2.2.3.1

Divide $- 6$ by $- 1$ .

$x = 6$

Step 1.2.2.4

The final solution is all the values that make $(x + 3) (6 - x) = 0$ true.

$x = - 3, 6$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{(x + 3) (6 - x)}{(x - 2) (x + 3)}$ true.

$x = 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \frac{((0) + 3) (6 - (0))}{((0) - 2) ((0) + 3)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{(0 + 3) (6 - (0))}{((0) - 2) ((0) + 3)}$

Step 2.2.2

Remove parentheses.

$y = \frac{(0 + 3) (6 - (0))}{(0 - 2) ((0) + 3)}$

Step 2.2.3

Remove parentheses.

$y = \frac{(0 + 3) (6 - (0))}{(0 - 2) (0 + 3)}$

Step 2.2.4

Remove parentheses.

$y = \frac{((0) + 3) (6 - (0))}{((0) - 2) ((0) + 3)}$

Step 2.2.5

Simplify $\frac{((0) + 3) (6 - (0))}{((0) - 2) ((0) + 3)}$ .

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Step 2.2.5.1

Cancel the common factor of $(0) + 3$ .

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Step 2.2.5.1.1

Cancel the common factor.

$y = \frac{((0) + 3) (6 - (0))}{((0) - 2) ((0) + 3)}$

Step 2.2.5.1.2

Rewrite the expression.

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

Step 2.2.5.2

Cancel the common factor of $6 - (0)$ and $(0) - 2$ .

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Step 2.2.5.2.1

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 2.2.5.2.2

Factor $- 1$ out of $- 1 (- 6) - (0)$ .

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

Step 2.2.5.2.3

Factor $2$ out of $- 1 (- 6 + 0)$ .

$y = \frac{2 (- 1 (- 3 + 0))}{0 - 2}$

Step 2.2.5.2.4

Cancel the common factors.

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Step 2.2.5.2.4.1

Factor $2$ out of $0$ .

$y = \frac{2 (- 1 (- 3 + 0))}{2 (0) - 2}$

Step 2.2.5.2.4.2

Factor $2$ out of $- 2$ .

$y = \frac{2 (- 1 (- 3 + 0))}{2 \cdot 0 + 2 \cdot - 1}$

Step 2.2.5.2.4.3

Factor $2$ out of $2 \cdot 0 + 2 \cdot - 1$ .

$y = \frac{2 (- 1 (- 3 + 0))}{2 \cdot (0 - 1)}$

Step 2.2.5.2.4.4

Cancel the common factor.

$y = \frac{2 (- 1 (- 3 + 0))}{2 \cdot (0 - 1)}$

Step 2.2.5.2.4.5

Rewrite the expression.

$y = \frac{- 1 (- 3 + 0)}{0 - 1}$

Step 2.2.5.3

Simplify the expression.

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Step 2.2.5.3.1

Add $- 3$ and $0$ .

$y = \frac{- 1 \cdot - 3}{0 - 1}$

Step 2.2.5.3.2

Subtract $1$ from $0$ .

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

Step 2.2.5.3.3

Multiply $- 1$ by $- 3$ .

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

Step 2.2.5.3.4

Divide $3$ by $- 1$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(6, 0)$

y-intercept(s): $(0, - 3)$

Step 4