Precalculus Examples

$y = x \sqrt{4 - x^{2}}$

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 $x$ .

$0 = x \sqrt{4 - x^{2}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x \sqrt{4 - x^{2}} = 0$ .

$x \sqrt{4 - x^{2}} = 0$

Step 1.2.2

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

${(x \sqrt{4 - x^{2}})}^{2} = 0^{2}$

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 $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

${(x {(4 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${(x {(4 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.1

Apply the product rule to $x {(4 - x^{2})}^{\frac{1}{2}}$ .

$x^{2} {({(4 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.3.2.1.2

Multiply the exponents in ${({(4 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.1.2.1

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

$x^{2} {(4 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

$x^{2} {(4 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.3.2.1.2.2.2

Rewrite the expression.

$x^{2} {(4 - x^{2})}^{1} = 0^{2}$

Step 1.2.3.2.1.3

Simplify.

$x^{2} (4 - x^{2}) = 0^{2}$

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.

$x^{2} \cdot 4 + x^{2} (- x^{2}) = 0^{2}$

Step 1.2.3.2.1.4.2

Reorder.

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Step 1.2.3.2.1.4.2.1

Move $4$ to the left of $x^{2}$ .

$4 \cdot x^{2} + x^{2} (- x^{2}) = 0^{2}$

Step 1.2.3.2.1.4.2.2

Rewrite using the commutative property of multiplication.

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

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 $x^{2}$ .

$4 x^{2} - (x^{2} x^{2}) = 0^{2}$

Step 1.2.3.2.1.5.2

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

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

Step 1.2.3.2.1.5.3

Add $2$ and $2$ .

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

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 $0$ .

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

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Factor the left side of the equation.

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Step 1.2.4.1.1

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

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Step 1.2.4.1.1.1

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

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

Step 1.2.4.1.1.2

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

$x^{2} \cdot 4 + x^{2} (- x^{2}) = 0$

Step 1.2.4.1.1.3

Factor $x^{2}$ out of $x^{2} \cdot 4 + x^{2} (- x^{2})$ .

$x^{2} (4 - x^{2}) = 0$

Step 1.2.4.1.2

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

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

Step 1.2.4.1.3

Factor.

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Step 1.2.4.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

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

Step 1.2.4.1.3.2

Remove unnecessary parentheses.

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

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 $0$ .

$x^{2} = 0$

$2 + x = 0$

$2 - x = 0$

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 $2 + x$ equal to $0$ and solve for $x$ .

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Step 1.2.4.4.1

Set $2 + x$ equal to $0$ .

$2 + x = 0$

Step 1.2.4.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.4.5

Set $2 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.4.5.1

Set $2 - x$ equal to $0$ .

$2 - x = 0$

Step 1.2.4.5.2

Solve $2 - x = 0$ for $x$ .

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Step 1.2.4.5.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 1.2.4.5.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 1.2.4.5.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

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

Step 1.2.4.5.2.2.2

Simplify the left side.

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Step 1.2.4.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.4.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.4.5.2.2.3

Simplify the right side.

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Step 1.2.4.5.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 1.2.4.6

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

$x = 0, - 2, 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 2, 0), (2, 0)$

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 $y$ .

$y = (0) \sqrt{4 - {(0)}^{2}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 \sqrt{4 - 0^{2}}$

Step 2.2.2

Remove parentheses.

$y = (0) \sqrt{4 - {(0)}^{2}}$

Step 2.2.3

Simplify $(0) \sqrt{4 - {(0)}^{2}}$ .

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Step 2.2.3.1

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

$y = 0 \sqrt{4 - 0}$

Step 2.2.3.2

Multiply $- 1$ by $0$ .

$y = 0 \sqrt{4 + 0}$

Step 2.2.3.3

Add $4$ and $0$ .

$y = 0 \sqrt{4}$

Step 2.2.3.4

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

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

Step 2.2.3.5

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

$y = 0 \cdot 2$

Step 2.2.3.6

Multiply $0$ by $2$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 2, 0), (2, 0)$

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

Step 4