Calculus Examples

$f (x) = x^{3} - 18 x^{2} + 81 x$

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 $x$ .

$0 = x^{3} - 18 x^{2} + 81 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{3} - 18 x^{2} + 81 x = 0$ .

$x^{3} - 18 x^{2} + 81 x = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $x$ out of $x^{3} - 18 x^{2} + 81 x$ .

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Step 1.2.2.1.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 18 x^{2} + 81 x = 0$

Step 1.2.2.1.2

Factor $x$ out of $- 18 x^{2}$ .

$x \cdot x^{2} + x (- 18 x) + 81 x = 0$

Step 1.2.2.1.3

Factor $x$ out of $81 x$ .

$x \cdot x^{2} + x (- 18 x) + x \cdot 81 = 0$

Step 1.2.2.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 18 x)$ .

$x (x^{2} - 18 x) + x \cdot 81 = 0$

Step 1.2.2.1.5

Factor $x$ out of $x (x^{2} - 18 x) + x \cdot 81$ .

$x (x^{2} - 18 x + 81) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

Rewrite $81$ as $9^{2}$ .

$x (x^{2} - 18 x + 9^{2}) = 0$

Step 1.2.2.2.2

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

$18 x = 2 \cdot x \cdot 9$

Step 1.2.2.2.3

Rewrite the polynomial.

$x (x^{2} - 2 \cdot x \cdot 9 + 9^{2}) = 0$

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 9$ .

$x {(x - 9)}^{2} = 0$

Step 1.2.3

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

$x = 0$

${(x - 9)}^{2} = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set ${(x - 9)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set ${(x - 9)}^{2}$ equal to $0$ .

${(x - 9)}^{2} = 0$

Step 1.2.5.2

Solve ${(x - 9)}^{2} = 0$ for $x$ .

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Step 1.2.5.2.1

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 1.2.5.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 1.2.6

The final solution is all the values that make $x {(x - 9)}^{2} = 0$ true.

$x = 0, 9$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (9, 0)$

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 $y$ .

$y = {(0)}^{3} - 18 {(0)}^{2} + 81 (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{3} - 18 {(0)}^{2} + 81 (0)$

Step 2.2.2

Remove parentheses.

$y = 0^{3} - 18 \cdot 0^{2} + 81 (0)$

Step 2.2.3

Remove parentheses.

$y = {(0)}^{3} - 18 {(0)}^{2} + 81 (0)$

Step 2.2.4

Simplify ${(0)}^{3} - 18 {(0)}^{2} + 81 (0)$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = 0 - 18 {(0)}^{2} + 81 (0)$

Step 2.2.4.1.2

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

$y = 0 - 18 \cdot 0 + 81 (0)$

Step 2.2.4.1.3

Multiply $- 18$ by $0$ .

$y = 0 + 0 + 81 (0)$

Step 2.2.4.1.4

Multiply $81$ by $0$ .

$y = 0 + 0 + 0$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (9, 0)$

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

Step 4