Pre-Algebra Examples

$x^{2} - y^{2} = 21$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Divide each term by $21$ to make the right side equal to one.

$\frac{x^{2}}{21} - \frac{y^{2}}{21} = \frac{21}{21}$

Step 1.2

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{x^{2}}{21} - \frac{y^{2}}{21} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = \sqrt{21}$

$b = \sqrt{21}$

$k = 0$

$h = 0$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the hyperbola by using the following formula.

$\sqrt{a^{2} + b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(\sqrt{21})}^{2} + {(\sqrt{21})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 5.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\sqrt{{(21^{\frac{1}{2}})}^{2} + {(\sqrt{21})}^{2}}$

Step 5.3.1.2

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

$\sqrt{21^{\frac{1}{2} \cdot 2} + {(\sqrt{21})}^{2}}$

Step 5.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{21^{\frac{2}{2}} + {(\sqrt{21})}^{2}}$

Step 5.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.1.4.1

Cancel the common factor.

$\sqrt{21^{\frac{2}{2}} + {(\sqrt{21})}^{2}}$

Step 5.3.1.4.2

Rewrite the expression.

$\sqrt{21^{1} + {(\sqrt{21})}^{2}}$

Step 5.3.1.5

Evaluate the exponent.

$\sqrt{21 + {(\sqrt{21})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\sqrt{21 + {(21^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{21 + 21^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{21 + 21^{\frac{2}{2}}}$

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

$\sqrt{21 + 21^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{21 + 21^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{21 + 21}$

Step 5.3.3

Add $21$ and $21$ .

$\sqrt{42}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(\sqrt{21}, 0)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.4

Substitute the known values of $h$ , $a$ , and $k$ into the formula and simplify.

$(- \sqrt{21}, 0)$

Step 6.5

The vertices of a hyperbola follow the form of $(h \pm a, k)$ . Hyperbolas have two vertices.

$(\sqrt{21}, 0), (- \sqrt{21}, 0)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(\sqrt{42}, 0)$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.4

Substitute the known values of $h$ , $c$ , and $k$ into the formula and simplify.

$(- \sqrt{42}, 0)$

Step 7.5

The foci of a hyperbola follow the form of $(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(\sqrt{42}, 0), (- \sqrt{42}, 0)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(\sqrt{21})}^{2} + {(\sqrt{21})}^{2}}}{\sqrt{21}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 8.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{\sqrt{{(21^{\frac{1}{2}})}^{2} + {\sqrt{21}}^{2}}}{\sqrt{21}}$

Step 8.3.1.1.2

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

$\frac{\sqrt{21^{\frac{1}{2} \cdot 2} + {\sqrt{21}}^{2}}}{\sqrt{21}}$

Step 8.3.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{21^{\frac{2}{2}} + {\sqrt{21}}^{2}}}{\sqrt{21}}$

Step 8.3.1.1.4

Cancel the common factor of $2$ .

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Step 8.3.1.1.4.1

Cancel the common factor.

$\frac{\sqrt{21^{\frac{2}{2}} + {\sqrt{21}}^{2}}}{\sqrt{21}}$

Step 8.3.1.1.4.2

Rewrite the expression.

$\frac{\sqrt{21^{1} + {\sqrt{21}}^{2}}}{\sqrt{21}}$

Step 8.3.1.1.5

Evaluate the exponent.

$\frac{\sqrt{21 + {\sqrt{21}}^{2}}}{\sqrt{21}}$

Step 8.3.1.2

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 8.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{\sqrt{21 + {(21^{\frac{1}{2}})}^{2}}}{\sqrt{21}}$

Step 8.3.1.2.2

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

$\frac{\sqrt{21 + 21^{\frac{1}{2} \cdot 2}}}{\sqrt{21}}$

Step 8.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{21 + 21^{\frac{2}{2}}}}{\sqrt{21}}$

Step 8.3.1.2.4

Cancel the common factor of $2$ .

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Step 8.3.1.2.4.1

Cancel the common factor.

$\frac{\sqrt{21 + 21^{\frac{2}{2}}}}{\sqrt{21}}$

Step 8.3.1.2.4.2

Rewrite the expression.

$\frac{\sqrt{21 + 21^{1}}}{\sqrt{21}}$

Step 8.3.1.2.5

Evaluate the exponent.

$\frac{\sqrt{21 + 21}}{\sqrt{21}}$

Step 8.3.1.3

Add $21$ and $21$ .

$\frac{\sqrt{42}}{\sqrt{21}}$

Step 8.3.2

Combine $\sqrt{42}$ and $\sqrt{21}$ into a single radical.

$\sqrt{\frac{42}{21}}$

Step 8.3.3

Divide $42$ by $21$ .

$\sqrt{2}$

Step 9

Find the focal parameter.

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Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

$\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{{\sqrt{21}}^{2}}{\sqrt{42}}$

Step 9.3

Simplify.

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Step 9.3.1

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 9.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{{(21^{\frac{1}{2}})}^{2}}{\sqrt{42}}$

Step 9.3.1.2

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

$\frac{21^{\frac{1}{2} \cdot 2}}{\sqrt{42}}$

Step 9.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{21^{\frac{2}{2}}}{\sqrt{42}}$

Step 9.3.1.4

Cancel the common factor of $2$ .

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Step 9.3.1.4.1

Cancel the common factor.

$\frac{21^{\frac{2}{2}}}{\sqrt{42}}$

Step 9.3.1.4.2

Rewrite the expression.

$\frac{21^{1}}{\sqrt{42}}$

Step 9.3.1.5

Evaluate the exponent.

$\frac{21}{\sqrt{42}}$

Step 9.3.2

Multiply $\frac{21}{\sqrt{42}}$ by $\frac{\sqrt{42}}{\sqrt{42}}$ .

$\frac{21}{\sqrt{42}} \cdot \frac{\sqrt{42}}{\sqrt{42}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{21}{\sqrt{42}}$ by $\frac{\sqrt{42}}{\sqrt{42}}$ .

$\frac{21 \sqrt{42}}{\sqrt{42} \sqrt{42}}$

Step 9.3.3.2

Raise $\sqrt{42}$ to the power of $1$ .

$\frac{21 \sqrt{42}}{{\sqrt{42}}^{1} \sqrt{42}}$

Step 9.3.3.3

Raise $\sqrt{42}$ to the power of $1$ .

$\frac{21 \sqrt{42}}{{\sqrt{42}}^{1} {\sqrt{42}}^{1}}$

Step 9.3.3.4

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

$\frac{21 \sqrt{42}}{{\sqrt{42}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{21 \sqrt{42}}{{\sqrt{42}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{42}}^{2}$ as $42$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{42}$ as $42^{\frac{1}{2}}$ .

$\frac{21 \sqrt{42}}{{(42^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{21 \sqrt{42}}{42^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{21 \sqrt{42}}{42^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\frac{21 \sqrt{42}}{42^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{21 \sqrt{42}}{42^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{21 \sqrt{42}}{42}$

Step 9.3.4

Cancel the common factor of $21$ and $42$ .

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Step 9.3.4.1

Factor $21$ out of $21 \sqrt{42}$ .

$\frac{21 (\sqrt{42})}{42}$

Step 9.3.4.2

Cancel the common factors.

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Step 9.3.4.2.1

Factor $21$ out of $42$ .

$\frac{21 \sqrt{42}}{21 \cdot 2}$

Step 9.3.4.2.2

Cancel the common factor.

$\frac{21 \sqrt{42}}{21 \cdot 2}$

Step 9.3.4.2.3

Rewrite the expression.

$\frac{\sqrt{42}}{2}$

Step 10

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm 1 \cdot x + 0$

Step 11

Simplify $1 \cdot x + 0$ .

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Step 11.1

Add $1 \cdot x$ and $0$ .

$y = 1 \cdot x$

Step 11.2

Multiply $x$ by $1$ .

$y = x$

Step 12

Simplify $- 1 \cdot x + 0$ .

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Step 12.1

Add $- 1 \cdot x$ and $0$ .

$y = - 1 \cdot x$

Step 12.2

Rewrite $- 1 x$ as $- x$ .

$y = - x$

Step 13

This hyperbola has two asymptotes.

$y = x, y = - x$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(0, 0)$

Vertices: $(\sqrt{21}, 0), (- \sqrt{21}, 0)$

Foci: $(\sqrt{42}, 0), (- \sqrt{42}, 0)$

Eccentricity: $\sqrt{2}$

Focal Parameter: $\frac{\sqrt{42}}{2}$

Asymptotes: $y = x$ , $y = - x$

Step 15