Precalculus Examples

$\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$

Step 1

Subtract $\frac{x^{2}}{16}$ from both sides of the equation.

$\frac{y^{2}}{4} = 1 - \frac{x^{2}}{16}$

Step 2

Multiply both sides of the equation by $4$ .

$4 \frac{y^{2}}{4} = 4 (1 - \frac{x^{2}}{16})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $4$ .

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Step 3.1.1.1

Cancel the common factor.

$4 \frac{y^{2}}{4} = 4 (1 - \frac{x^{2}}{16})$

Step 3.1.1.2

Rewrite the expression.

$y^{2} = 4 (1 - \frac{x^{2}}{16})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $4 (1 - \frac{x^{2}}{16})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = 4 \cdot 1 + 4 (- \frac{x^{2}}{16})$

Step 3.2.1.2

Multiply $4$ by $1$ .

$y^{2} = 4 + 4 (- \frac{x^{2}}{16})$

Step 3.2.1.3

Cancel the common factor of $4$ .

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Step 3.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{16}$ into the numerator.

$y^{2} = 4 + 4 \frac{- x^{2}}{16}$

Step 3.2.1.3.2

Factor $4$ out of $16$ .

$y^{2} = 4 + 4 \frac{- x^{2}}{4 (4)}$

Step 3.2.1.3.3

Cancel the common factor.

$y^{2} = 4 + 4 \frac{- x^{2}}{4 \cdot 4}$

Step 3.2.1.3.4

Rewrite the expression.

$y^{2} = 4 + \frac{- x^{2}}{4}$

Step 3.2.1.4

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify $\pm \sqrt{4 - \frac{x^{2}}{4}}$ .

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Step 5.1

Write the expression using exponents.

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Step 5.1.1

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

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

Step 5.1.2

Rewrite $\frac{x^{2}}{4}$ as ${(\frac{x}{2})}^{2}$ .

$y = \pm \sqrt{2^{2} - {(\frac{x}{2})}^{2}}$

Step 5.2

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 $b = \frac{x}{2}$ .

$y = \pm \sqrt{(2 + \frac{x}{2}) (2 - \frac{x}{2})}$

Step 5.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \pm \sqrt{(2 \cdot \frac{2}{2} + \frac{x}{2}) (2 - \frac{x}{2})}$

Step 5.4

Combine $2$ and $\frac{2}{2}$ .

$y = \pm \sqrt{(\frac{2 \cdot 2}{2} + \frac{x}{2}) (2 - \frac{x}{2})}$

Step 5.5

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{2 \cdot 2 + x}{2} (2 - \frac{x}{2})}$

Step 5.6

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{4 + x}{2} (2 - \frac{x}{2})}$

Step 5.7

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \pm \sqrt{\frac{4 + x}{2} (2 \cdot \frac{2}{2} - \frac{x}{2})}$

Step 5.8

Combine $2$ and $\frac{2}{2}$ .

$y = \pm \sqrt{\frac{4 + x}{2} (\frac{2 \cdot 2}{2} - \frac{x}{2})}$

Step 5.9

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 + x}{2} \cdot \frac{2 \cdot 2 - x}{2}}$

Step 5.10

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{4 + x}{2} \cdot \frac{4 - x}{2}}$

Step 5.11

Multiply $\frac{4 + x}{2}$ by $\frac{4 - x}{2}$ .

$y = \pm \sqrt{\frac{(4 + x) (4 - x)}{2 \cdot 2}}$

Step 5.12

Multiply $2$ by $2$ .

$y = \pm \sqrt{\frac{(4 + x) (4 - x)}{4}}$

Step 5.13

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

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Step 5.13.1

Factor the perfect power $1^{2}$ out of $(4 + x) (4 - x)$ .

$y = \pm \sqrt{\frac{1^{2} ((4 + x) (4 - x))}{4}}$

Step 5.13.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{\frac{1^{2} ((4 + x) (4 - x))}{2^{2} \cdot 1}}$

Step 5.13.3

Rearrange the fraction $\frac{1^{2} ((4 + x) (4 - x))}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} ((4 + x) (4 - x))}$

Step 5.14

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{(4 + x) (4 - x)}$

Step 5.15

Combine $\frac{1}{2}$ and $\sqrt{(4 + x) (4 - x)}$ .

$y = \pm \frac{\sqrt{(4 + x) (4 - x)}}{2}$

Step 6

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

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Step 6.1

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

$y = \frac{\sqrt{(4 + x) (4 - x)}}{2}$

Step 6.2

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

$y = - \frac{\sqrt{(4 + x) (4 - x)}}{2}$

Step 6.3

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

$y = \frac{\sqrt{(4 + x) (4 - x)}}{2}$

$y = - \frac{\sqrt{(4 + x) (4 - x)}}{2}$

$y = \frac{\sqrt{(4 + x) (4 - x)}}{2}$

$y = - \frac{\sqrt{(4 + x) (4 - x)}}{2}$

Step 7

Set the radicand in $\sqrt{(4 + x) (4 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(4 + x) (4 - x) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

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

$4 + x = 0$

$4 - x = 0$

Step 8.2

Set $4 + x$ equal to $0$ and solve for $x$ .

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Step 8.2.1

Set $4 + x$ equal to $0$ .

$4 + x = 0$

Step 8.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 8.3

Set $4 - x$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Step 8.3.2

Solve $4 - x = 0$ for $x$ .

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Step 8.3.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 8.3.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 8.3.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 8.3.2.2.2

Simplify the left side.

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Step 8.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 8.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 8.3.2.2.3

Simplify the right side.

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Step 8.3.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 8.4

The final solution is all the values that make $(4 + x) (4 - x) \geq 0$ true.

$x = - 4, 4$

Step 8.5

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 4$

$x > 4$

Step 8.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.6.1

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 8.6.1.2

Replace $x$ with $- 6$ in the original inequality.

$(4 - 6) (4 - (- 6)) \geq 0$

Step 8.6.1.3

The left side $- 20$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.2

Test a value on the interval $- 4 < x < 4$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 4 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 8.6.2.2

Replace $x$ with $0$ in the original inequality.

$(4 + 0) (4 - (0)) \geq 0$

Step 8.6.2.3

The left side $16$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 8.6.3.2

Replace $x$ with $6$ in the original inequality.

$(4 + 6) (4 - (6)) \geq 0$

Step 8.6.3.3

The left side $- 20$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 4$ False

$- 4 < x < 4$ True

$x > 4$ False

$x < - 4$ False

$- 4 < x < 4$ True

$x > 4$ False

Step 8.7

The solution consists of all of the true intervals.

$- 4 \leq x \leq 4$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 4, 4]$

Set-Builder Notation:

${x | - 4 \leq x \leq 4}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- 2, 2]$

Set-Builder Notation:

${y | - 2 \leq y \leq 2}$

Step 11

Determine the domain and range.

Domain: $[- 4, 4], {x | - 4 \leq x \leq 4}$

Range: $[- 2, 2], {y | - 2 \leq y \leq 2}$

Step 12