Precalculus Examples

$\frac{y^{2}}{9} - \frac{x^{2}}{25} = 1$

Step 1

Add $\frac{x^{2}}{25}$ to both sides of the equation.

$\frac{y^{2}}{9} = 1 + \frac{x^{2}}{25}$

Step 2

Multiply both sides of the equation by $9$ .

$9 \frac{y^{2}}{9} = 9 (1 + \frac{x^{2}}{25})$

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 $9$ .

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Step 3.1.1.1

Cancel the common factor.

$9 \frac{y^{2}}{9} = 9 (1 + \frac{x^{2}}{25})$

Step 3.1.1.2

Rewrite the expression.

$y^{2} = 9 (1 + \frac{x^{2}}{25})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $9 (1 + \frac{x^{2}}{25})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = 9 \cdot 1 + 9 \frac{x^{2}}{25}$

Step 3.2.1.2

Multiply $9$ by $1$ .

$y^{2} = 9 + 9 \frac{x^{2}}{25}$

Step 3.2.1.3

Combine $9$ and $\frac{x^{2}}{25}$ .

$y^{2} = 9 + \frac{9 x^{2}}{25}$

Step 4

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

$y = \pm \sqrt{9 + \frac{9 x^{2}}{25}}$

Step 5

Simplify $\pm \sqrt{9 + \frac{9 x^{2}}{25}}$ .

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Step 5.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = \pm \sqrt{9 \cdot \frac{25}{25} + \frac{9 x^{2}}{25}}$

Step 5.2

Combine $9$ and $\frac{25}{25}$ .

$y = \pm \sqrt{\frac{9 \cdot 25}{25} + \frac{9 x^{2}}{25}}$

Step 5.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{9 \cdot 25 + 9 x^{2}}{25}}$

Step 5.4

Factor $9$ out of $9 \cdot 25 + 9 x^{2}$ .

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Step 5.4.1

Factor $9$ out of $9 \cdot 25$ .

$y = \pm \sqrt{\frac{9 (25) + 9 x^{2}}{25}}$

Step 5.4.2

Factor $9$ out of $9 x^{2}$ .

$y = \pm \sqrt{\frac{9 (25) + 9 (x^{2})}{25}}$

Step 5.4.3

Factor $9$ out of $9 (25) + 9 (x^{2})$ .

$y = \pm \sqrt{\frac{9 (25 + x^{2})}{25}}$

Step 5.5

Rewrite $\frac{9 (25 + x^{2})}{25}$ as ${(\frac{3}{5})}^{2} (5^{2} + x^{2})$ .

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Step 5.5.1

Factor the perfect power $3^{2}$ out of $9 (25 + x^{2})$ .

$y = \pm \sqrt{\frac{3^{2} (5^{2} + x^{2})}{25}}$

Step 5.5.2

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

$y = \pm \sqrt{\frac{3^{2} (5^{2} + x^{2})}{5^{2} \cdot 1}}$

Step 5.5.3

Rearrange the fraction $\frac{3^{2} (5^{2} + x^{2})}{5^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{3}{5})}^{2} (5^{2} + x^{2})}$

Step 5.6

Pull terms out from under the radical.

$y = \pm \frac{3}{5} \sqrt{5^{2} + x^{2}}$

Step 5.7

Raise $5$ to the power of $2$ .

$y = \pm \frac{3}{5} \sqrt{25 + x^{2}}$

Step 5.8

Combine $\frac{3}{5}$ and $\sqrt{25 + x^{2}}$ .

$y = \pm \frac{3 \sqrt{25 + x^{2}}}{5}$

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.

$y = \frac{3 \sqrt{25 + x^{2}}}{5}$

Step 6.2

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

$y = - \frac{3 \sqrt{25 + x^{2}}}{5}$

Step 6.3

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

$y = \frac{3 \sqrt{25 + x^{2}}}{5}$

$y = - \frac{3 \sqrt{25 + x^{2}}}{5}$

$y = \frac{3 \sqrt{25 + x^{2}}}{5}$

$y = - \frac{3 \sqrt{25 + x^{2}}}{5}$

Step 7

Set the radicand in $\sqrt{25 + x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$25 + x^{2} \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

Subtract $25$ from both sides of the inequality.

$x^{2} \geq - 25$

Step 8.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 9

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 10

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

Interval Notation:

$(- \infty, - 3] \cup [3, \infty)$

Set-Builder Notation:

${y | y \leq - 3, y \geq 3}$

Step 11

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $(- \infty, - 3] \cup [3, \infty), {y | y \leq - 3, y \geq 3}$

Step 12