Calculus Examples

$f (x) = \frac{x^{2} - 6 x + 9}{x^{2} - x - 6}$

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 $x$ .

$0 = \frac{x^{2} - 6 x + 9}{x^{2} - x - 6}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 6 x + 9 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor using the perfect square rule.

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Step 1.2.2.1.1

Rewrite $9$ as $3^{2}$ .

$x^{2} - 6 x + 3^{2} = 0$

Step 1.2.2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.2.1.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 3 + 3^{2} = 0$

Step 1.2.2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

${(x - 3)}^{2} = 0$

Step 1.2.2.2

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

$x - 3 = 0$

Step 1.2.2.3

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{2} - 6 x + 9}{x^{2} - x - 6}$ true.

$No solution$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = \frac{{(0)}^{2} - 6 \cdot 0 + 9}{{(0)}^{2} - (0) - 6}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 6 \cdot 0 + 9}{{(0)}^{2} - (0) - 6}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 6 \cdot 0 + 9}{0^{2} - (0) - 6}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 6 \cdot 0 + 9}{{(0)}^{2} - (0) - 6}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 6 \cdot 0 + 9}{{(0)}^{2} - (0) - 6}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{0 - 6 \cdot 0 + 9}{0^{2} - (0) - 6}$

Step 2.2.4.1.2

Multiply $- 6$ by $0$ .

$y = \frac{0 + 0 + 9}{0^{2} - (0) - 6}$

Step 2.2.4.1.3

Add $0$ and $0$ .

$y = \frac{0 + 9}{0^{2} - (0) - 6}$

Step 2.2.4.1.4

Add $0$ and $9$ .

$y = \frac{9}{0^{2} - (0) - 6}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{9}{0 - (0) - 6}$

Step 2.2.4.2.2

Multiply $- 1$ by $0$ .

$y = \frac{9}{0 + 0 - 6}$

Step 2.2.4.2.3

Add $0$ and $0$ .

$y = \frac{9}{0 - 6}$

Step 2.2.4.2.4

Subtract $6$ from $0$ .

$y = \frac{9}{- 6}$

Step 2.2.4.3

Reduce the expression by cancelling the common factors.

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Step 2.2.4.3.1

Cancel the common factor of $9$ and $- 6$ .

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Step 2.2.4.3.1.1

Factor $3$ out of $9$ .

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

Step 2.2.4.3.1.2

Cancel the common factors.

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Step 2.2.4.3.1.2.1

Factor $3$ out of $- 6$ .

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

Step 2.2.4.3.1.2.2

Cancel the common factor.

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

Step 2.2.4.3.1.2.3

Rewrite the expression.

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

Step 2.2.4.3.2

Move the negative in front of the fraction.

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3}{2})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{3}{2})$

Step 4