Algebra Examples

$f (x) = \frac{(x - 5) (x - 12)}{(x + 3) (x - 2) (x - 5)}$

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 $x$ .

$0 = \frac{(x - 5) (x - 12)}{(x + 3) (x - 2) (x - 5)}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$(x - 5) (x - 12) = 0$

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 $0$ .

$x - 5 = 0$

$x - 12 = 0$

Step 1.2.2.2

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.2.2.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 1.2.2.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.2.3

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 1.2.2.3.2

Add $12$ to both sides of the equation.

$x = 12$

Step 1.2.2.4

The final solution is all the values that make $(x - 5) (x - 12) = 0$ true.

$x = 5, 12$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{(x - 5) (x - 12)}{(x + 3) (x - 2) (x - 5)}$ true.

$x = 12$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(12, 0)$

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 $y$ .

$y = \frac{((0) - 5) ((0) - 12)}{((0) + 3) ((0) - 2) ((0) - 5)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{(0 - 5) ((0) - 12)}{((0) + 3) ((0) - 2) ((0) - 5)}$

Step 2.2.2

Remove parentheses.

$y = \frac{(0 - 5) (0 - 12)}{((0) + 3) ((0) - 2) ((0) - 5)}$

Step 2.2.3

Remove parentheses.

$y = \frac{(0 - 5) (0 - 12)}{(0 + 3) ((0) - 2) ((0) - 5)}$

Step 2.2.4

Remove parentheses.

$y = \frac{(0 - 5) (0 - 12)}{(0 + 3) (0 - 2) ((0) - 5)}$

Step 2.2.5

Remove parentheses.

$y = \frac{(0 - 5) (0 - 12)}{(0 + 3) (0 - 2) (0 - 5)}$

Step 2.2.6

Remove parentheses.

$y = \frac{((0) - 5) ((0) - 12)}{((0) + 3) ((0) - 2) ((0) - 5)}$

Step 2.2.7

Simplify $\frac{((0) - 5) ((0) - 12)}{((0) + 3) ((0) - 2) ((0) - 5)}$ .

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Step 2.2.7.1

Reduce the expression by cancelling the common factors.

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Step 2.2.7.1.1

Cancel the common factor of $(0) - 5$ .

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Step 2.2.7.1.1.1

Cancel the common factor.

$y = \frac{((0) - 5) ((0) - 12)}{((0) + 3) ((0) - 2) ((0) - 5)}$

Step 2.2.7.1.1.2

Rewrite the expression.

$y = \frac{(0) - 12}{((0) + 3) ((0) - 2)}$

Step 2.2.7.1.2

Cancel the common factor of $(0) - 12$ and $(0) + 3$ .

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Step 2.2.7.1.2.1

Factor $3$ out of $0$ .

$y = \frac{3 (0) - 12}{((0) + 3) ((0) - 2)}$

Step 2.2.7.1.2.2

Factor $3$ out of $- 12$ .

$y = \frac{3 \cdot 0 + 3 \cdot - 4}{((0) + 3) ((0) - 2)}$

Step 2.2.7.1.2.3

Factor $3$ out of $3 \cdot 0 + 3 \cdot - 4$ .

$y = \frac{3 \cdot (0 - 4)}{((0) + 3) ((0) - 2)}$

Step 2.2.7.1.2.4

Cancel the common factors.

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Step 2.2.7.1.2.4.1

Factor $3$ out of $((0) + 3) ((0) - 2)$ .

$y = \frac{3 \cdot (0 - 4)}{3 ((0 + 1) ((0) - 2))}$

Step 2.2.7.1.2.4.2

Cancel the common factor.

$y = \frac{3 \cdot (0 - 4)}{3 ((0 + 1) ((0) - 2))}$

Step 2.2.7.1.2.4.3

Rewrite the expression.

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

Step 2.2.7.1.3

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

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Step 2.2.7.1.3.1

Factor $2$ out of $0$ .

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

Step 2.2.7.1.3.2

Factor $2$ out of $- 4$ .

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

Step 2.2.7.1.3.3

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

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

Step 2.2.7.1.3.4

Cancel the common factors.

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Step 2.2.7.1.3.4.1

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

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

Step 2.2.7.1.3.4.2

Cancel the common factor.

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

Step 2.2.7.1.3.4.3

Rewrite the expression.

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

Step 2.2.7.1.4

Subtract $2$ from $0$ .

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

Step 2.2.7.2

Simplify the denominator.

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Step 2.2.7.2.1

Rewrite $0$ as $- 1 (0)$ .

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

Step 2.2.7.2.2

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

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

Step 2.2.7.2.3

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

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

Step 2.2.7.2.4

Raise $0 - 1$ to the power of $1$ .

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

Step 2.2.7.2.5

Raise $0 - 1$ to the power of $1$ .

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

Step 2.2.7.2.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.7.2.7

Add $1$ and $1$ .

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

Step 2.2.7.3

Simplify the denominator.

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Step 2.2.7.3.1

Subtract $1$ from $0$ .

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

Step 2.2.7.3.2

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

$y = \frac{- 2}{- 1 \cdot 1}$

Step 2.2.7.4

Simplify the expression.

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Step 2.2.7.4.1

Multiply $- 1$ by $1$ .

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

Step 2.2.7.4.2

Divide $- 2$ by $- 1$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(12, 0)$

y-intercept(s): $(0, 2)$

Step 4