Calculus Examples

$x^{2} - 9 y^{2} = 200$

Step 1

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

$- 9 y^{2} = 200 - x^{2}$

Step 2

Divide each term in $- 9 y^{2} = 200 - x^{2}$ by $- 9$ and simplify.

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Step 2.1

Divide each term in $- 9 y^{2} = 200 - x^{2}$ by $- 9$ .

$\frac{- 9 y^{2}}{- 9} = \frac{200}{- 9} + \frac{- x^{2}}{- 9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 9$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 9 y^{2}}{- 9} = \frac{200}{- 9} + \frac{- x^{2}}{- 9}$

Step 2.2.1.2

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

$y^{2} = \frac{200}{- 9} + \frac{- x^{2}}{- 9}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{200}{9} + \frac{- x^{2}}{- 9}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = - \frac{200}{9} + \frac{x^{2}}{9}$

Step 3

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

$y = \pm \sqrt{- \frac{200}{9} + \frac{x^{2}}{9}}$

Step 4

Simplify $\pm \sqrt{- \frac{200}{9} + \frac{x^{2}}{9}}$ .

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Step 4.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- 200 + x^{2}}{9}}$

Step 4.2

Rewrite $\frac{- 200 + x^{2}}{9}$ as ${(\frac{1}{3})}^{2} (- 200 + x^{2})$ .

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Step 4.2.1

Factor the perfect power $1^{2}$ out of $- 200 + x^{2}$ .

$y = \pm \sqrt{\frac{1^{2} (- 200 + x^{2})}{9}}$

Step 4.2.2

Factor the perfect power $3^{2}$ out of $9$ .

$y = \pm \sqrt{\frac{1^{2} (- 200 + x^{2})}{3^{2} \cdot 1}}$

Step 4.2.3

Rearrange the fraction $\frac{1^{2} (- 200 + x^{2})}{3^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{3})}^{2} (- 200 + x^{2})}$

Step 4.3

Pull terms out from under the radical.

$y = \pm \frac{1}{3} \sqrt{- 200 + x^{2}}$

Step 4.4

Combine $\frac{1}{3}$ and $\sqrt{- 200 + x^{2}}$ .

$y = \pm \frac{\sqrt{- 200 + x^{2}}}{3}$

Step 5

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

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Step 5.1

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

$y = \frac{\sqrt{- 200 + x^{2}}}{3}$

Step 5.2

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

$y = - \frac{\sqrt{- 200 + x^{2}}}{3}$

Step 5.3

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

$y = \frac{\sqrt{- 200 + x^{2}}}{3}$

$y = - \frac{\sqrt{- 200 + x^{2}}}{3}$

$y = \frac{\sqrt{- 200 + x^{2}}}{3}$

$y = - \frac{\sqrt{- 200 + x^{2}}}{3}$

Step 6

Set the radicand in $\sqrt{- 200 + x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$- 200 + x^{2} \geq 0$

Step 7

Solve for $x$ .

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Step 7.1

Add $200$ to both sides of the inequality.

$x^{2} \geq 200$

Step 7.2

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

$\sqrt{x^{2}} \geq \sqrt{200}$

Step 7.3

Simplify the equation.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Pull terms out from under the radical.

$| x | \geq \sqrt{200}$

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

Simplify $\sqrt{200}$ .

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Step 7.3.2.1.1

Rewrite $200$ as $10^{2} \cdot 2$ .

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Step 7.3.2.1.1.1

Factor $100$ out of $200$ .

$| x | \geq \sqrt{100 (2)}$

Step 7.3.2.1.1.2

Rewrite $100$ as $10^{2}$ .

$| x | \geq \sqrt{10^{2} \cdot 2}$

Step 7.3.2.1.2

Pull terms out from under the radical.

$| x | \geq | 10 | \sqrt{2}$

Step 7.3.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$| x | \geq 10 \sqrt{2}$

Step 7.4

Write $| x | \geq 10 \sqrt{2}$ as a piecewise.

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Step 7.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 7.4.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \geq 10 \sqrt{2}$

Step 7.4.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 7.4.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq 10 \sqrt{2}$

Step 7.4.5

Write as a piecewise.

${\begin{cases} x \geq 10 \sqrt{2} & x \geq 0 \\ - x \geq 10 \sqrt{2} & x < 0 \end{cases}$

Step 7.5

Find the intersection of $x \geq 10 \sqrt{2}$ and $x \geq 0$ .

$x \geq 10 \sqrt{2}$

Step 7.6

Divide each term in $- x \geq 10 \sqrt{2}$ by $- 1$ and simplify.

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Step 7.6.1

Divide each term in $- x \geq 10 \sqrt{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{10 \sqrt{2}}{- 1}$

Step 7.6.2

Simplify the left side.

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Step 7.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{10 \sqrt{2}}{- 1}$

Step 7.6.2.2

Divide $x$ by $1$ .

$x \leq \frac{10 \sqrt{2}}{- 1}$

Step 7.6.3

Simplify the right side.

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Step 7.6.3.1

Move the negative one from the denominator of $\frac{10 \sqrt{2}}{- 1}$ .

$x \leq - 1 \cdot (10 \sqrt{2})$

Step 7.6.3.2

Rewrite $- 1 \cdot (10 \sqrt{2})$ as $- (10 \sqrt{2})$ .

$x \leq - (10 \sqrt{2})$

Step 7.6.3.3

Multiply $10$ by $- 1$ .

$x \leq - 10 \sqrt{2}$

Step 7.7

Find the union of the solutions.

$x \leq - 10 \sqrt{2}$ or $x \geq 10 \sqrt{2}$

Step 8

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

Interval Notation:

$(- \infty, - 10 \sqrt{2}] \cup [10 \sqrt{2}, \infty)$

Set-Builder Notation:

${x | x \leq - 10 \sqrt{2}, x \geq 10 \sqrt{2}}$

Step 9

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 10

Determine the domain and range.

Domain: $(- \infty, - 10 \sqrt{2}] \cup [10 \sqrt{2}, \infty), {x | x \leq - 10 \sqrt{2}, x \geq 10 \sqrt{2}}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 11