Precalculus Examples

$x^{2} + 2 x + 15 y = - 3 y^{2} + 7$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Move all terms containing variables to the left side of the equation.

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Step 1.1.1

Add $3 y^{2}$ to both sides of the equation.

$x^{2} + 2 x + 15 y + 3 y^{2} = 7$

Step 1.1.2

Move $15 y$ .

$x^{2} + 2 x + 3 y^{2} + 15 y = 7$

Step 1.1.3

Move $2 x$ .

$x^{2} + 3 y^{2} + 2 x + 15 y = 7$

Step 1.2

Complete the square for $x^{2} + 2 x$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 2$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot 1}$

Step 1.2.3.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1

Cancel the common factor.

$d = \frac{2}{2 \cdot 1}$

Step 1.2.3.2.2

Rewrite the expression.

$d = 1$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{2^{2}}{4 \cdot 1}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

$e = 0 - \frac{4}{4 \cdot 1}$

Step 1.2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{4}{4}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ .

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Step 1.2.4.2.1.3.1

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Step 1.2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 1.2.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 1.2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 1)}^{2} - 1$ .

${(x + 1)}^{2} - 1$

Step 1.3

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + 3 y^{2} + 2 x + 15 y = 7$ .

${(x + 1)}^{2} - 1 + 3 y^{2} + 15 y = 7$

Step 1.4

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

${(x + 1)}^{2} + 3 y^{2} + 15 y = 7 + 1$

Step 1.5

Complete the square for $3 y^{2} + 15 y$ .

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Step 1.5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 3$

$b = 15$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{15}{2 \cdot 3}$

Step 1.5.3.2

Cancel the common factor of $15$ and $3$ .

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Step 1.5.3.2.1

Factor $3$ out of $15$ .

$d = \frac{3 \cdot 5}{2 \cdot 3}$

Step 1.5.3.2.2

Cancel the common factors.

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Step 1.5.3.2.2.1

Factor $3$ out of $2 \cdot 3$ .

$d = \frac{3 \cdot 5}{3 \cdot 2}$

Step 1.5.3.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot 5}{3 \cdot 2}$

Step 1.5.3.2.2.3

Rewrite the expression.

$d = \frac{5}{2}$

Step 1.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{15^{2}}{4 \cdot 3}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Raise $15$ to the power of $2$ .

$e = 0 - \frac{225}{4 \cdot 3}$

Step 1.5.4.2.1.2

Multiply $4$ by $3$ .

$e = 0 - \frac{225}{12}$

Step 1.5.4.2.1.3

Cancel the common factor of $225$ and $12$ .

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Step 1.5.4.2.1.3.1

Factor $3$ out of $225$ .

$e = 0 - \frac{3 (75)}{12}$

Step 1.5.4.2.1.3.2

Cancel the common factors.

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Step 1.5.4.2.1.3.2.1

Factor $3$ out of $12$ .

$e = 0 - \frac{3 \cdot 75}{3 \cdot 4}$

Step 1.5.4.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{3 \cdot 75}{3 \cdot 4}$

Step 1.5.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{75}{4}$

Step 1.5.4.2.2

Subtract $\frac{75}{4}$ from $0$ .

$e = - \frac{75}{4}$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y + \frac{5}{2})}^{2} - \frac{75}{4}$ .

$3 {(y + \frac{5}{2})}^{2} - \frac{75}{4}$

Step 1.6

Substitute $3 {(y + \frac{5}{2})}^{2} - \frac{75}{4}$ for $3 y^{2} + 15 y$ in the equation $x^{2} + 3 y^{2} + 2 x + 15 y = 7$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} - \frac{75}{4} = 7 + 1$

Step 1.7

Move $- \frac{75}{4}$ to the right side of the equation by adding $\frac{75}{4}$ to both sides.

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = 7 + 1 + \frac{75}{4}$

Step 1.8

Simplify $7 + 1 + \frac{75}{4}$ .

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Step 1.8.1

Find the common denominator.

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Step 1.8.1.1

Write $7$ as a fraction with denominator $1$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7}{1} + 1 + \frac{75}{4}$

Step 1.8.1.2

Multiply $\frac{7}{1}$ by $\frac{4}{4}$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7}{1} \cdot \frac{4}{4} + 1 + \frac{75}{4}$

Step 1.8.1.3

Multiply $\frac{7}{1}$ by $\frac{4}{4}$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7 \cdot 4}{4} + 1 + \frac{75}{4}$

Step 1.8.1.4

Write $1$ as a fraction with denominator $1$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7 \cdot 4}{4} + \frac{1}{1} + \frac{75}{4}$

Step 1.8.1.5

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7 \cdot 4}{4} + \frac{1}{1} \cdot \frac{4}{4} + \frac{75}{4}$

Step 1.8.1.6

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7 \cdot 4}{4} + \frac{4}{4} + \frac{75}{4}$

Step 1.8.2

Combine the numerators over the common denominator.

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{7 \cdot 4 + 4 + 75}{4}$

Step 1.8.3

Simplify the expression.

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Step 1.8.3.1

Multiply $7$ by $4$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{28 + 4 + 75}{4}$

Step 1.8.3.2

Add $28$ and $4$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{32 + 75}{4}$

Step 1.8.3.3

Add $32$ and $75$ .

${(x + 1)}^{2} + 3 {(y + \frac{5}{2})}^{2} = \frac{107}{4}$

Step 1.9

Divide each term by $\frac{107}{4}$ to make the right side equal to one.

$\frac{{(x + 1)}^{2}}{\frac{107}{4}} + \frac{3 {(y + \frac{5}{2})}^{2}}{\frac{107}{4}} = \frac{\frac{107}{4}}{\frac{107}{4}}$

Step 1.10

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 $1$ .

$\frac{{(x + 1)}^{2}}{\frac{107}{4}} + \frac{{(y + \frac{5}{2})}^{2}}{\frac{107}{12}} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = \frac{\sqrt{107}}{2}$

$b = \frac{\sqrt{321}}{6}$

$k = - \frac{5}{2}$

$h = - 1$

Step 4

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

$(- 1, - \frac{5}{2})$

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 ellipse by using the following formula.

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

Step 5.2

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

$\sqrt{{(\frac{\sqrt{107}}{2})}^{2} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Apply the product rule to $\frac{\sqrt{107}}{2}$ .

$\sqrt{\frac{{\sqrt{107}}^{2}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{107}}^{2}$ as $107$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{107}$ as $107^{\frac{1}{2}}$ .

$\sqrt{\frac{{(107^{\frac{1}{2}})}^{2}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.2.2

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

$\sqrt{\frac{107^{\frac{1}{2} \cdot 2}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.2.3

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

$\sqrt{\frac{107^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

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.

$\sqrt{\frac{107^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{\frac{107^{1}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{\frac{107}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

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

$\sqrt{\frac{107}{4} - {(\frac{\sqrt{321}}{6})}^{2}}$

Step 5.3.3.2

Apply the product rule to $\frac{\sqrt{321}}{6}$ .

$\sqrt{\frac{107}{4} - \frac{{\sqrt{321}}^{2}}{6^{2}}}$

Step 5.3.4

Rewrite ${\sqrt{321}}^{2}$ as $321$ .

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{321}$ as $321^{\frac{1}{2}}$ .

$\sqrt{\frac{107}{4} - \frac{{(321^{\frac{1}{2}})}^{2}}{6^{2}}}$

Step 5.3.4.2

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

$\sqrt{\frac{107}{4} - \frac{321^{\frac{1}{2} \cdot 2}}{6^{2}}}$

Step 5.3.4.3

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

$\sqrt{\frac{107}{4} - \frac{321^{\frac{2}{2}}}{6^{2}}}$

Step 5.3.4.4

Cancel the common factor of $2$ .

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Step 5.3.4.4.1

Cancel the common factor.

$\sqrt{\frac{107}{4} - \frac{321^{\frac{2}{2}}}{6^{2}}}$

Step 5.3.4.4.2

Rewrite the expression.

$\sqrt{\frac{107}{4} - \frac{321^{1}}{6^{2}}}$

Step 5.3.4.5

Evaluate the exponent.

$\sqrt{\frac{107}{4} - \frac{321}{6^{2}}}$

Step 5.3.5

Raise $6$ to the power of $2$ .

$\sqrt{\frac{107}{4} - \frac{321}{36}}$

Step 5.3.6

Cancel the common factor of $321$ and $36$ .

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Step 5.3.6.1

Factor $3$ out of $321$ .

$\sqrt{\frac{107}{4} - \frac{3 (107)}{36}}$

Step 5.3.6.2

Cancel the common factors.

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Step 5.3.6.2.1

Factor $3$ out of $36$ .

$\sqrt{\frac{107}{4} - \frac{3 \cdot 107}{3 \cdot 12}}$

Step 5.3.6.2.2

Cancel the common factor.

$\sqrt{\frac{107}{4} - \frac{3 \cdot 107}{3 \cdot 12}}$

Step 5.3.6.2.3

Rewrite the expression.

$\sqrt{\frac{107}{4} - \frac{107}{12}}$

Step 5.3.7

To write $\frac{107}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{107}{4} \cdot \frac{3}{3} - \frac{107}{12}}$

Step 5.3.8

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.8.1

Multiply $\frac{107}{4}$ by $\frac{3}{3}$ .

$\sqrt{\frac{107 \cdot 3}{4 \cdot 3} - \frac{107}{12}}$

Step 5.3.8.2

Multiply $4$ by $3$ .

$\sqrt{\frac{107 \cdot 3}{12} - \frac{107}{12}}$

Step 5.3.9

Combine the numerators over the common denominator.

$\sqrt{\frac{107 \cdot 3 - 107}{12}}$

Step 5.3.10

Simplify the numerator.

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Step 5.3.10.1

Multiply $107$ by $3$ .

$\sqrt{\frac{321 - 107}{12}}$

Step 5.3.10.2

Subtract $107$ from $321$ .

$\sqrt{\frac{214}{12}}$

Step 5.3.11

Cancel the common factor of $214$ and $12$ .

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Step 5.3.11.1

Factor $2$ out of $214$ .

$\sqrt{\frac{2 (107)}{12}}$

Step 5.3.11.2

Cancel the common factors.

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Step 5.3.11.2.1

Factor $2$ out of $12$ .

$\sqrt{\frac{2 \cdot 107}{2 \cdot 6}}$

Step 5.3.11.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 107}{2 \cdot 6}}$

Step 5.3.11.2.3

Rewrite the expression.

$\sqrt{\frac{107}{6}}$

Step 5.3.12

Rewrite $\sqrt{\frac{107}{6}}$ as $\frac{\sqrt{107}}{\sqrt{6}}$ .

$\frac{\sqrt{107}}{\sqrt{6}}$

Step 5.3.13

Multiply $\frac{\sqrt{107}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{107}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 5.3.14

Combine and simplify the denominator.

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Step 5.3.14.1

Multiply $\frac{\sqrt{107}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{107} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 5.3.14.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{107} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 5.3.14.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{107} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 5.3.14.4

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

$\frac{\sqrt{107} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 5.3.14.5

Add $1$ and $1$ .

$\frac{\sqrt{107} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 5.3.14.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.3.14.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{107} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 5.3.14.6.2

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

$\frac{\sqrt{107} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 5.3.14.6.3

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

$\frac{\sqrt{107} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 5.3.14.6.4

Cancel the common factor of $2$ .

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Step 5.3.14.6.4.1

Cancel the common factor.

$\frac{\sqrt{107} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 5.3.14.6.4.2

Rewrite the expression.

$\frac{\sqrt{107} \sqrt{6}}{6^{1}}$

Step 5.3.14.6.5

Evaluate the exponent.

$\frac{\sqrt{107} \sqrt{6}}{6}$

Step 5.3.15

Simplify the numerator.

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Step 5.3.15.1

Combine using the product rule for radicals.

$\frac{\sqrt{107 \cdot 6}}{6}$

Step 5.3.15.2

Multiply $107$ by $6$ .

$\frac{\sqrt{642}}{6}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse 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.

$(- 1 + \frac{\sqrt{107}}{2}, - \frac{5}{2})$

Step 6.3

The second vertex of an ellipse 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.

$(- 1 - (\frac{\sqrt{107}}{2}), - \frac{5}{2})$

Step 6.5

Simplify.

$(- 1 - \frac{\sqrt{107}}{2}, - \frac{5}{2})$

Step 6.6

Ellipses have two vertices.

${Vertex}_{1}$ : $(- 1 + \frac{\sqrt{107}}{2}, - \frac{5}{2})$

${Vertex}_{2}$ : $(- 1 - \frac{\sqrt{107}}{2}, - \frac{5}{2})$

${Vertex}_{1}$ : $(- 1 + \frac{\sqrt{107}}{2}, - \frac{5}{2})$

${Vertex}_{2}$ : $(- 1 - \frac{\sqrt{107}}{2}, - \frac{5}{2})$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse 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.

$(- 1 + \frac{\sqrt{642}}{6}, - \frac{5}{2})$

Step 7.3

The second focus of an ellipse 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.

$(- 1 - (\frac{\sqrt{642}}{6}), - \frac{5}{2})$

Step 7.5

Simplify.

$(- 1 - \frac{\sqrt{642}}{6}, - \frac{5}{2})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(- 1 + \frac{\sqrt{642}}{6}, - \frac{5}{2})$

${Focus}_{2}$ : $(- 1 - \frac{\sqrt{642}}{6}, - \frac{5}{2})$

${Focus}_{1}$ : $(- 1 + \frac{\sqrt{642}}{6}, - \frac{5}{2})$

${Focus}_{2}$ : $(- 1 - \frac{\sqrt{642}}{6}, - \frac{5}{2})$

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.

$\frac{\sqrt{{(\frac{\sqrt{107}}{2})}^{2} - {(\frac{\sqrt{321}}{6})}^{2}}}{\frac{\sqrt{107}}{2}}$

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{\sqrt{107}}{2})}^{2} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.2

Apply the product rule to $\frac{\sqrt{107}}{2}$ .

$\sqrt{\frac{{\sqrt{107}}^{2}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.3

Rewrite ${\sqrt{107}}^{2}$ as $107$ .

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Step 8.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{107}$ as $107^{\frac{1}{2}}$ .

$\sqrt{\frac{{(107^{\frac{1}{2}})}^{2}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.3.2

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

$\sqrt{\frac{107^{\frac{1}{2} \cdot 2}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.3.3

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

$\sqrt{\frac{107^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.3.4

Cancel the common factor of $2$ .

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Step 8.3.3.4.1

Cancel the common factor.

$\sqrt{\frac{107^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.3.4.2

Rewrite the expression.

$\sqrt{\frac{107^{1}}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.3.5

Evaluate the exponent.

$\sqrt{\frac{107}{2^{2}} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.4

Simplify the expression.

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Step 8.3.4.1

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

$\sqrt{\frac{107}{4} - {(\frac{\sqrt{321}}{6})}^{2}} \frac{2}{\sqrt{107}}$

Step 8.3.4.2

Apply the product rule to $\frac{\sqrt{321}}{6}$ .

$\sqrt{\frac{107}{4} - \frac{{\sqrt{321}}^{2}}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.5

Rewrite ${\sqrt{321}}^{2}$ as $321$ .

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Step 8.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{321}$ as $321^{\frac{1}{2}}$ .

$\sqrt{\frac{107}{4} - \frac{{(321^{\frac{1}{2}})}^{2}}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.5.2

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

$\sqrt{\frac{107}{4} - \frac{321^{\frac{1}{2} \cdot 2}}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.5.3

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

$\sqrt{\frac{107}{4} - \frac{321^{\frac{2}{2}}}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.5.4

Cancel the common factor of $2$ .

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Step 8.3.5.4.1

Cancel the common factor.

$\sqrt{\frac{107}{4} - \frac{321^{\frac{2}{2}}}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.5.4.2

Rewrite the expression.

$\sqrt{\frac{107}{4} - \frac{321^{1}}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.5.5

Evaluate the exponent.

$\sqrt{\frac{107}{4} - \frac{321}{6^{2}}} \frac{2}{\sqrt{107}}$

Step 8.3.6

Raise $6$ to the power of $2$ .

$\sqrt{\frac{107}{4} - \frac{321}{36}} \frac{2}{\sqrt{107}}$

Step 8.3.7

Cancel the common factor of $321$ and $36$ .

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Step 8.3.7.1

Factor $3$ out of $321$ .

$\sqrt{\frac{107}{4} - \frac{3 (107)}{36}} \frac{2}{\sqrt{107}}$

Step 8.3.7.2

Cancel the common factors.

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Step 8.3.7.2.1

Factor $3$ out of $36$ .

$\sqrt{\frac{107}{4} - \frac{3 \cdot 107}{3 \cdot 12}} \frac{2}{\sqrt{107}}$

Step 8.3.7.2.2

Cancel the common factor.

$\sqrt{\frac{107}{4} - \frac{3 \cdot 107}{3 \cdot 12}} \frac{2}{\sqrt{107}}$

Step 8.3.7.2.3

Rewrite the expression.

$\sqrt{\frac{107}{4} - \frac{107}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.8

To write $\frac{107}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{107}{4} \cdot \frac{3}{3} - \frac{107}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.9

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.9.1

Multiply $\frac{107}{4}$ by $\frac{3}{3}$ .

$\sqrt{\frac{107 \cdot 3}{4 \cdot 3} - \frac{107}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.9.2

Multiply $4$ by $3$ .

$\sqrt{\frac{107 \cdot 3}{12} - \frac{107}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.10

Combine the numerators over the common denominator.

$\sqrt{\frac{107 \cdot 3 - 107}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.11

Simplify the numerator.

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Step 8.3.11.1

Multiply $107$ by $3$ .

$\sqrt{\frac{321 - 107}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.11.2

Subtract $107$ from $321$ .

$\sqrt{\frac{214}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.12

Cancel the common factor of $214$ and $12$ .

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Step 8.3.12.1

Factor $2$ out of $214$ .

$\sqrt{\frac{2 (107)}{12}} \frac{2}{\sqrt{107}}$

Step 8.3.12.2

Cancel the common factors.

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Step 8.3.12.2.1

Factor $2$ out of $12$ .

$\sqrt{\frac{2 \cdot 107}{2 \cdot 6}} \frac{2}{\sqrt{107}}$

Step 8.3.12.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 107}{2 \cdot 6}} \frac{2}{\sqrt{107}}$

Step 8.3.12.2.3

Rewrite the expression.

$\sqrt{\frac{107}{6}} \frac{2}{\sqrt{107}}$

Step 8.3.13

Rewrite $\sqrt{\frac{107}{6}}$ as $\frac{\sqrt{107}}{\sqrt{6}}$ .

$\frac{\sqrt{107}}{\sqrt{6}} \cdot \frac{2}{\sqrt{107}}$

Step 8.3.14

Cancel the common factor of $\sqrt{107}$ .

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Step 8.3.14.1

Cancel the common factor.

$\frac{\sqrt{107}}{\sqrt{6}} \cdot \frac{2}{\sqrt{107}}$

Step 8.3.14.2

Rewrite the expression.

$\frac{1}{\sqrt{6}} \cdot 2$

Step 8.3.15

Combine $\frac{1}{\sqrt{6}}$ and $2$ .

$\frac{2}{\sqrt{6}}$

Step 8.3.16

Multiply $\frac{2}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 8.3.17

Combine and simplify the denominator.

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Step 8.3.17.1

Multiply $\frac{2}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2 \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 8.3.17.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 8.3.17.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 8.3.17.4

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

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 8.3.17.5

Add $1$ and $1$ .

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 8.3.17.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 8.3.17.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{2 \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 8.3.17.6.2

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

$\frac{2 \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 8.3.17.6.3

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

$\frac{2 \sqrt{6}}{6^{\frac{2}{2}}}$

Step 8.3.17.6.4

Cancel the common factor of $2$ .

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Step 8.3.17.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{6}}{6^{\frac{2}{2}}}$

Step 8.3.17.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{6}}{6^{1}}$

Step 8.3.17.6.5

Evaluate the exponent.

$\frac{2 \sqrt{6}}{6}$

Step 8.3.18

Cancel the common factor of $2$ and $6$ .

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Step 8.3.18.1

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{2 (\sqrt{6})}{6}$

Step 8.3.18.2

Cancel the common factors.

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Step 8.3.18.2.1

Factor $2$ out of $6$ .

$\frac{2 \sqrt{6}}{2 \cdot 3}$

Step 8.3.18.2.2

Cancel the common factor.

$\frac{2 \sqrt{6}}{2 \cdot 3}$

Step 8.3.18.2.3

Rewrite the expression.

$\frac{\sqrt{6}}{3}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(- 1, - \frac{5}{2})$

${Vertex}_{1}$ : $(- 1 + \frac{\sqrt{107}}{2}, - \frac{5}{2})$

${Vertex}_{2}$ : $(- 1 - \frac{\sqrt{107}}{2}, - \frac{5}{2})$

${Focus}_{1}$ : $(- 1 + \frac{\sqrt{642}}{6}, - \frac{5}{2})$

${Focus}_{2}$ : $(- 1 - \frac{\sqrt{642}}{6}, - \frac{5}{2})$

Eccentricity: $\frac{\sqrt{6}}{3}$

Step 10