Calculus Examples

$y = x \sqrt{5 - x^{2}}$

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 $x$ .

$0 = x \sqrt{5 - x^{2}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x \sqrt{5 - x^{2}} = 0$ .

$x \sqrt{5 - x^{2}} = 0$

Step 1.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(x \sqrt{5 - x^{2}})}^{2} = 0^{2}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - x^{2}}$ as ${(5 - x^{2})}^{\frac{1}{2}}$ .

${(x {(5 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${(x {(5 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1

Apply the product rule to $x {(5 - x^{2})}^{\frac{1}{2}}$ .

$x^{2} {({(5 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.3.2.1.2

Multiply the exponents in ${({(5 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2} {(5 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.3.2.1.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.2.2.1

Cancel the common factor.

$x^{2} {(5 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.3.2.1.2.2.2

Rewrite the expression.

$x^{2} {(5 - x^{2})}^{1} = 0^{2}$

Step 1.2.3.2.1.3

Simplify.

$x^{2} (5 - x^{2}) = 0^{2}$

Step 1.2.3.2.1.4

Simplify by multiplying through.

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Step 1.2.3.2.1.4.1

Apply the distributive property.

$x^{2} \cdot 5 + x^{2} (- x^{2}) = 0^{2}$

Step 1.2.3.2.1.4.2

Reorder.

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Step 1.2.3.2.1.4.2.1

Move $5$ to the left of $x^{2}$ .

$5 \cdot x^{2} + x^{2} (- x^{2}) = 0^{2}$

Step 1.2.3.2.1.4.2.2

Rewrite using the commutative property of multiplication.

$5 \cdot x^{2} - x^{2} x^{2} = 0^{2}$

Step 1.2.3.2.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.3.2.1.5.1

Move $x^{2}$ .

$5 x^{2} - (x^{2} x^{2}) = 0^{2}$

Step 1.2.3.2.1.5.2

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

$5 x^{2} - x^{2 + 2} = 0^{2}$

Step 1.2.3.2.1.5.3

Add $2$ and $2$ .

$5 x^{2} - x^{4} = 0^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

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

$5 x^{2} - x^{4} = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Factor $x^{2}$ out of $5 x^{2} - x^{4}$ .

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Step 1.2.4.1.1

Factor $x^{2}$ out of $5 x^{2}$ .

$x^{2} \cdot 5 - x^{4} = 0$

Step 1.2.4.1.2

Factor $x^{2}$ out of $- x^{4}$ .

$x^{2} \cdot 5 + x^{2} (- x^{2}) = 0$

Step 1.2.4.1.3

Factor $x^{2}$ out of $x^{2} \cdot 5 + x^{2} (- x^{2})$ .

$x^{2} (5 - x^{2}) = 0$

Step 1.2.4.2

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

$x^{2} = 0$

$5 - x^{2} = 0$

Step 1.2.4.3

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

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Step 1.2.4.3.1

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

$x^{2} = 0$

Step 1.2.4.3.2

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

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Step 1.2.4.3.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.4.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.4.3.2.2.2

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

$x = \pm 0$

Step 1.2.4.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4.4

Set $5 - x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.4.1

Set $5 - x^{2}$ equal to $0$ .

$5 - x^{2} = 0$

Step 1.2.4.4.2

Solve $5 - x^{2} = 0$ for $x$ .

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Step 1.2.4.4.2.1

Subtract $5$ from both sides of the equation.

$- x^{2} = - 5$

Step 1.2.4.4.2.2

Divide each term in $- x^{2} = - 5$ by $- 1$ and simplify.

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Step 1.2.4.4.2.2.1

Divide each term in $- x^{2} = - 5$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 5}{- 1}$

Step 1.2.4.4.2.2.2

Simplify the left side.

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Step 1.2.4.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 5}{- 1}$

Step 1.2.4.4.2.2.2.2

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

$x^{2} = \frac{- 5}{- 1}$

Step 1.2.4.4.2.2.3

Simplify the right side.

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Step 1.2.4.4.2.2.3.1

Divide $- 5$ by $- 1$ .

$x^{2} = 5$

Step 1.2.4.4.2.3

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

$x = \pm \sqrt{5}$

Step 1.2.4.4.2.4

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

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Step 1.2.4.4.2.4.1

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

$x = \sqrt{5}$

Step 1.2.4.4.2.4.2

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

$x = - \sqrt{5}$

Step 1.2.4.4.2.4.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 1.2.4.5

The final solution is all the values that make $x^{2} (5 - x^{2}) = 0$ true.

$x = 0, \sqrt{5}, - \sqrt{5}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0)$

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 $y$ .

$y = (0) \sqrt{5 - {(0)}^{2}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 \sqrt{5 - 0^{2}}$

Step 2.2.2

Remove parentheses.

$y = (0) \sqrt{5 - {(0)}^{2}}$

Step 2.2.3

Simplify $(0) \sqrt{5 - {(0)}^{2}}$ .

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Step 2.2.3.1

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

$y = 0 \sqrt{5 - 0}$

Step 2.2.3.2

Multiply $- 1$ by $0$ .

$y = 0 \sqrt{5 + 0}$

Step 2.2.3.3

Add $5$ and $0$ .

$y = 0 \sqrt{5}$

Step 2.2.3.4

Multiply $0$ by $\sqrt{5}$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0)$

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

Step 4