Precalculus Examples

$\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1$

Step 1

Subtract $\frac{x^{2}}{100}$ from both sides of the equation.

$\frac{y^{2}}{64} = 1 - \frac{x^{2}}{100}$

Step 2

Multiply both sides of the equation by $64$ .

$64 \frac{y^{2}}{64} = 64 (1 - \frac{x^{2}}{100})$

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 $64$ .

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Step 3.1.1.1

Cancel the common factor.

$64 \frac{y^{2}}{64} = 64 (1 - \frac{x^{2}}{100})$

Step 3.1.1.2

Rewrite the expression.

$y^{2} = 64 (1 - \frac{x^{2}}{100})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $64 (1 - \frac{x^{2}}{100})$ .

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Step 3.2.1.1

Simplify terms.

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Step 3.2.1.1.1

Apply the distributive property.

$y^{2} = 64 \cdot 1 + 64 (- \frac{x^{2}}{100})$

Step 3.2.1.1.2

Multiply $64$ by $1$ .

$y^{2} = 64 + 64 (- \frac{x^{2}}{100})$

Step 3.2.1.1.3

Cancel the common factor of $4$ .

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Step 3.2.1.1.3.1

Move the leading negative in $- \frac{x^{2}}{100}$ into the numerator.

$y^{2} = 64 + 64 \frac{- x^{2}}{100}$

Step 3.2.1.1.3.2

Factor $4$ out of $64$ .

$y^{2} = 64 + 4 (16) \frac{- x^{2}}{100}$

Step 3.2.1.1.3.3

Factor $4$ out of $100$ .

$y^{2} = 64 + 4 \cdot 16 \frac{- x^{2}}{4 \cdot 25}$

Step 3.2.1.1.3.4

Cancel the common factor.

$y^{2} = 64 + 4 \cdot 16 \frac{- x^{2}}{4 \cdot 25}$

Step 3.2.1.1.3.5

Rewrite the expression.

$y^{2} = 64 + 16 \frac{- x^{2}}{25}$

Step 3.2.1.1.4

Combine $16$ and $\frac{- x^{2}}{25}$ .

$y^{2} = 64 + \frac{16 (- x^{2})}{25}$

Step 3.2.1.1.5

Multiply $- 1$ by $16$ .

$y^{2} = 64 + \frac{- 16 x^{2}}{25}$

Step 3.2.1.2

Move the negative in front of the fraction.

$y^{2} = 64 - \frac{16 x^{2}}{25}$

Step 4

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

$y = \pm \sqrt{64 - \frac{16 x^{2}}{25}}$

Step 5

Simplify $\pm \sqrt{64 - \frac{16 x^{2}}{25}}$ .

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Step 5.1

Write the expression using exponents.

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Step 5.1.1

Rewrite $64$ as $8^{2}$ .

$y = \pm \sqrt{8^{2} - \frac{16 x^{2}}{25}}$

Step 5.1.2

Rewrite $\frac{16 x^{2}}{25}$ as ${(\frac{4 x}{5})}^{2}$ .

$y = \pm \sqrt{8^{2} - {(\frac{4 x}{5})}^{2}}$

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 $a = 8$ and $b = \frac{4 x}{5}$ .

$y = \pm \sqrt{(8 + \frac{4 x}{5}) (8 - \frac{4 x}{5})}$

Step 5.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \pm \sqrt{(8 \cdot \frac{5}{5} + \frac{4 x}{5}) (8 - \frac{4 x}{5})}$

Step 5.4

Simplify terms.

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Step 5.4.1

Combine $8$ and $\frac{5}{5}$ .

$y = \pm \sqrt{(\frac{8 \cdot 5}{5} + \frac{4 x}{5}) (8 - \frac{4 x}{5})}$

Step 5.4.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{8 \cdot 5 + 4 x}{5} (8 - \frac{4 x}{5})}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Factor $4$ out of $8 \cdot 5 + 4 x$ .

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Step 5.5.1.1

Factor $4$ out of $8 \cdot 5$ .

$y = \pm \sqrt{\frac{4 (2 \cdot 5) + 4 x}{5} (8 - \frac{4 x}{5})}$

Step 5.5.1.2

Factor $4$ out of $4 x$ .

$y = \pm \sqrt{\frac{4 (2 \cdot 5) + 4 (x)}{5} (8 - \frac{4 x}{5})}$

Step 5.5.1.3

Factor $4$ out of $4 (2 \cdot 5) + 4 (x)$ .

$y = \pm \sqrt{\frac{4 (2 \cdot 5 + x)}{5} (8 - \frac{4 x}{5})}$

Step 5.5.2

Multiply $2$ by $5$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} (8 - \frac{4 x}{5})}$

Step 5.6

To write $8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} (8 \cdot \frac{5}{5} - \frac{4 x}{5})}$

Step 5.7

Simplify terms.

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Step 5.7.1

Combine $8$ and $\frac{5}{5}$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} (\frac{8 \cdot 5}{5} - \frac{4 x}{5})}$

Step 5.7.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{4 (10 + x)}{5} \cdot \frac{8 \cdot 5 - 4 x}{5}}$

Step 5.8

Simplify the numerator.

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Step 5.8.1

Factor $4$ out of $8 \cdot 5 - 4 x$ .

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Step 5.8.1.1

Factor $4$ out of $8 \cdot 5$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} \cdot \frac{4 (2 \cdot 5) - 4 x}{5}}$

Step 5.8.1.2

Factor $4$ out of $- 4 x$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} \cdot \frac{4 (2 \cdot 5) + 4 (- x)}{5}}$

Step 5.8.1.3

Factor $4$ out of $4 (2 \cdot 5) + 4 (- x)$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} \cdot \frac{4 (2 \cdot 5 - x)}{5}}$

Step 5.8.2

Multiply $2$ by $5$ .

$y = \pm \sqrt{\frac{4 (10 + x)}{5} \cdot \frac{4 (10 - x)}{5}}$

Step 5.9

Combine fractions.

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Step 5.9.1

Multiply $\frac{4 (10 + x)}{5}$ by $\frac{4 (10 - x)}{5}$ .

$y = \pm \sqrt{\frac{4 (10 + x) (4 (10 - x))}{5 \cdot 5}}$

Step 5.9.2

Multiply.

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Step 5.9.2.1

Multiply $4$ by $4$ .

$y = \pm \sqrt{\frac{16 (10 + x) (10 - x)}{5 \cdot 5}}$

Step 5.9.2.2

Multiply $5$ by $5$ .

$y = \pm \sqrt{\frac{16 (10 + x) (10 - x)}{25}}$

Step 5.10

Rewrite $\frac{16 (10 + x) (10 - x)}{25}$ as ${(\frac{2^{2}}{5})}^{2} ((10 + x) (10 - x))$ .

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Step 5.10.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (10 + x) (10 - x)$ .

$y = \pm \sqrt{\frac{{(2^{2})}^{2} ((10 + x) (10 - x))}{25}}$

Step 5.10.2

Factor the perfect power $5^{2}$ out of $25$ .

$y = \pm \sqrt{\frac{{(2^{2})}^{2} ((10 + x) (10 - x))}{5^{2} \cdot 1}}$

Step 5.10.3

Rearrange the fraction $\frac{{(2^{2})}^{2} ((10 + x) (10 - x))}{5^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{2^{2}}{5})}^{2} ((10 + x) (10 - x))}$

Step 5.11

Pull terms out from under the radical.

$y = \pm \frac{2^{2}}{5} \sqrt{(10 + x) (10 - x)}$

Step 5.12

Raise $2$ to the power of $2$ .

$y = \pm \frac{4}{5} \sqrt{(10 + x) (10 - x)}$

Step 5.13

Combine $\frac{4}{5}$ and $\sqrt{(10 + x) (10 - x)}$ .

$y = \pm \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

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.

$y = \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

Step 6.2

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

$y = - \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

Step 6.3

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

$y = \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

$y = - \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

$y = \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

$y = - \frac{4 \sqrt{(10 + x) (10 - x)}}{5}$

Step 7

Set the radicand in $\sqrt{(10 + x) (10 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(10 + x) (10 - x) \geq 0$

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 $0$ .

$10 + x = 0$

$10 - x = 0$

Step 8.2

Set $10 + x$ equal to $0$ and solve for $x$ .

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Step 8.2.1

Set $10 + x$ equal to $0$ .

$10 + x = 0$

Step 8.2.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 8.3

Set $10 - x$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $10 - x$ equal to $0$ .

$10 - x = 0$

Step 8.3.2

Solve $10 - x = 0$ for $x$ .

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Step 8.3.2.1

Subtract $10$ from both sides of the equation.

$- x = - 10$

Step 8.3.2.2

Divide each term in $- x = - 10$ by $- 1$ and simplify.

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Step 8.3.2.2.1

Divide each term in $- x = - 10$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 10}{- 1}$

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.

$\frac{x}{1} = \frac{- 10}{- 1}$

Step 8.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 10}{- 1}$

Step 8.3.2.2.3

Simplify the right side.

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Step 8.3.2.2.3.1

Divide $- 10$ by $- 1$ .

$x = 10$

Step 8.4

The final solution is all the values that make $(10 + x) (10 - x) \geq 0$ true.

$x = - 10, 10$

Step 8.5

Use each root to create test intervals.

$x < - 10$

$- 10 < x < 10$

$x > 10$

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 $x < - 10$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 10$ and see if this value makes the original inequality true.

$x = - 12$

Step 8.6.1.2

Replace $x$ with $- 12$ in the original inequality.

$(10 - 12) (10 - (- 12)) \geq 0$

Step 8.6.1.3

The left side $- 44$ 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 $- 10 < x < 10$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 10 < x < 10$ 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.

$(10 + 0) (10 - (0)) \geq 0$

Step 8.6.2.3

The left side $100$ 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 $x > 10$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Step 8.6.3.2

Replace $x$ with $12$ in the original inequality.

$(10 + 12) (10 - (12)) \geq 0$

Step 8.6.3.3

The left side $- 44$ 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.

$x < - 10$ False

$- 10 < x < 10$ True

$x > 10$ False

$x < - 10$ False

$- 10 < x < 10$ True

$x > 10$ False

Step 8.7

The solution consists of all of the true intervals.

$- 10 \leq x \leq 10$

Step 9

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

Interval Notation:

$[- 10, 10]$

Set-Builder Notation:

${x | - 10 \leq x \leq 10}$

Step 10

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

Interval Notation:

$[- 8, 8]$

Set-Builder Notation:

${y | - 8 \leq y \leq 8}$

Step 11

Determine the domain and range.

Domain: $[- 10, 10], {x | - 10 \leq x \leq 10}$

Range: $[- 8, 8], {y | - 8 \leq y \leq 8}$

Step 12