Calculus Examples

$f (x) = (x^{2} + 12) (144 - x^{2})$

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 $x$ .

$0 = (x^{2} + 12) (144 - x^{2})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x^{2} + 12) (144 - x^{2}) = 0$ .

$(x^{2} + 12) (144 - x^{2}) = 0$

Step 1.2.2

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

$x^{2} + 12 = 0$

$144 - x^{2} = 0$

Step 1.2.3

Set $x^{2} + 12$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2} + 12$ equal to $0$ .

$x^{2} + 12 = 0$

Step 1.2.3.2

Solve $x^{2} + 12 = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $12$ from both sides of the equation.

$x^{2} = - 12$

Step 1.2.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 12}$

Step 1.2.3.2.3

Simplify $\pm \sqrt{- 12}$ .

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Step 1.2.3.2.3.1

Rewrite $- 12$ as $- 1 (12)$ .

$x = \pm \sqrt{- 1 (12)}$

Step 1.2.3.2.3.2

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{12}$

Step 1.2.3.2.3.3

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

$x = \pm i \cdot \sqrt{12}$

Step 1.2.3.2.3.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.3.2.3.4.1

Factor $4$ out of $12$ .

$x = \pm i \cdot \sqrt{4 (3)}$

Step 1.2.3.2.3.4.2

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2} \cdot 3}$

Step 1.2.3.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{3})$

Step 1.2.3.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i \sqrt{3}$

Step 1.2.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i \sqrt{3}$

Step 1.2.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i \sqrt{3}$

Step 1.2.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i \sqrt{3}, - 2 i \sqrt{3}$

Step 1.2.4

Set $144 - x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $144 - x^{2}$ equal to $0$ .

$144 - x^{2} = 0$

Step 1.2.4.2

Solve $144 - x^{2} = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $144$ from both sides of the equation.

$- x^{2} = - 144$

Step 1.2.4.2.2

Divide each term in $- x^{2} = - 144$ by $- 1$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $- x^{2} = - 144$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 144}{- 1}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 144}{- 1}$

Step 1.2.4.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 144}{- 1}$

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Divide $- 144$ by $- 1$ .

$x^{2} = 144$

Step 1.2.4.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{144}$

Step 1.2.4.2.4

Simplify $\pm \sqrt{144}$ .

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Step 1.2.4.2.4.1

Rewrite $144$ as $12^{2}$ .

$x = \pm \sqrt{12^{2}}$

Step 1.2.4.2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 12$

Step 1.2.4.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 12$

Step 1.2.4.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 12$

Step 1.2.4.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 12, - 12$

Step 1.2.5

The final solution is all the values that make $(x^{2} + 12) (144 - x^{2}) = 0$ true.

$x = 2 i \sqrt{3}, - 2 i \sqrt{3}, 12, - 12$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = ({(0)}^{2} + 12) (144 - {(0)}^{2})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0^{2} + 12) (144 - {(0)}^{2})$

Step 2.2.2

Remove parentheses.

$y = (0^{2} + 12) (144 - 0^{2})$

Step 2.2.3

Remove parentheses.

$y = ({(0)}^{2} + 12) (144 - {(0)}^{2})$

Step 2.2.4

Simplify $({(0)}^{2} + 12) (144 - {(0)}^{2})$ .

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Step 2.2.4.1

Simplify the expression.

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Step 2.2.4.1.1

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

$y = (0 + 12) (144 - {(0)}^{2})$

Step 2.2.4.1.2

Add $0$ and $12$ .

$y = 12 (144 - {(0)}^{2})$

Step 2.2.4.2

Simplify each term.

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Step 2.2.4.2.1

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

$y = 12 (144 - 0)$

Step 2.2.4.2.2

Multiply $- 1$ by $0$ .

$y = 12 (144 + 0)$

Step 2.2.4.3

Simplify the expression.

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Step 2.2.4.3.1

Add $144$ and $0$ .

$y = 12 \cdot 144$

Step 2.2.4.3.2

Multiply $12$ by $144$ .

$y = 1728$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1728)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 1728)$

Step 4