Calculus Examples

$y = \frac{x^{2} - 9 x + 36}{x - 5}$

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 $x$ .

$0 = \frac{x^{2} - 9 x + 36}{x - 5}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 9 x + 36 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.2.2

Substitute the values $a = 1$ , $b = - 9$ , and $c = 36$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot 36)}}{2 \cdot 1}$

Step 1.2.2.3

Simplify.

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Step 1.2.2.3.1

Simplify the numerator.

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Step 1.2.2.3.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}$

Step 1.2.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 36$ .

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Step 1.2.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 36}}{2 \cdot 1}$

Step 1.2.2.3.1.2.2

Multiply $- 4$ by $36$ .

$x = \frac{9 \pm \sqrt{81 - 144}}{2 \cdot 1}$

Step 1.2.2.3.1.3

Subtract $144$ from $81$ .

$x = \frac{9 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 1.2.2.3.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{9 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 1.2.2.3.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 1.2.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{9 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 1.2.2.3.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 1.2.2.3.1.7.1

Factor $9$ out of $63$ .

$x = \frac{9 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 1.2.2.3.1.7.2

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

$x = \frac{9 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 1.2.2.3.1.8

Pull terms out from under the radical.

$x = \frac{9 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 1.2.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{9 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 1.2.2.3.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm 3 i \sqrt{7}}{2}$

Step 1.2.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.2.4.1

Simplify the numerator.

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Step 1.2.2.4.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}$

Step 1.2.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 36$ .

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Step 1.2.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 36}}{2 \cdot 1}$

Step 1.2.2.4.1.2.2

Multiply $- 4$ by $36$ .

$x = \frac{9 \pm \sqrt{81 - 144}}{2 \cdot 1}$

Step 1.2.2.4.1.3

Subtract $144$ from $81$ .

$x = \frac{9 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 1.2.2.4.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{9 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 1.2.2.4.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 1.2.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{9 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 1.2.2.4.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 1.2.2.4.1.7.1

Factor $9$ out of $63$ .

$x = \frac{9 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 1.2.2.4.1.7.2

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

$x = \frac{9 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 1.2.2.4.1.8

Pull terms out from under the radical.

$x = \frac{9 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 1.2.2.4.1.9

Move $3$ to the left of $i$ .

$x = \frac{9 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 1.2.2.4.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm 3 i \sqrt{7}}{2}$

Step 1.2.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{9 + 3 i \sqrt{7}}{2}$

Step 1.2.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.2.5.1

Simplify the numerator.

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Step 1.2.2.5.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}$

Step 1.2.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 36$ .

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Step 1.2.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 36}}{2 \cdot 1}$

Step 1.2.2.5.1.2.2

Multiply $- 4$ by $36$ .

$x = \frac{9 \pm \sqrt{81 - 144}}{2 \cdot 1}$

Step 1.2.2.5.1.3

Subtract $144$ from $81$ .

$x = \frac{9 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 1.2.2.5.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{9 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 1.2.2.5.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 1.2.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{9 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 1.2.2.5.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 1.2.2.5.1.7.1

Factor $9$ out of $63$ .

$x = \frac{9 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 1.2.2.5.1.7.2

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

$x = \frac{9 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 1.2.2.5.1.8

Pull terms out from under the radical.

$x = \frac{9 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 1.2.2.5.1.9

Move $3$ to the left of $i$ .

$x = \frac{9 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 1.2.2.5.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm 3 i \sqrt{7}}{2}$

Step 1.2.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{9 - 3 i \sqrt{7}}{2}$

Step 1.2.2.6

The final answer is the combination of both solutions.

$x = \frac{9 + 3 i \sqrt{7}}{2}, \frac{9 - 3 i \sqrt{7}}{2}$

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 $y$ .

$y = \frac{{(0)}^{2} - 9 \cdot 0 + 36}{(0) - 5}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 9 \cdot 0 + 36}{(0) - 5}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 9 \cdot 0 + 36}{0 - 5}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 9 \cdot 0 + 36}{(0) - 5}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 9 \cdot 0 + 36}{(0) - 5}$ .

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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 $0$ .

$y = \frac{0 - 9 \cdot 0 + 36}{0 - 5}$

Step 2.2.4.1.2

Multiply $- 9$ by $0$ .

$y = \frac{0 + 0 + 36}{0 - 5}$

Step 2.2.4.1.3

Add $0$ and $0$ .

$y = \frac{0 + 36}{0 - 5}$

Step 2.2.4.1.4

Add $0$ and $36$ .

$y = \frac{36}{0 - 5}$

Step 2.2.4.2

Simplify the expression.

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Step 2.2.4.2.1

Subtract $5$ from $0$ .

$y = \frac{36}{- 5}$

Step 2.2.4.2.2

Move the negative in front of the fraction.

$y = - \frac{36}{5}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{36}{5})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{36}{5})$

Step 4