Pre-Algebra Examples

$18 x^{2} + 2 y^{2} - 126 x + 2 y + 203 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $203$ from both sides of the equation.

$18 x^{2} + 2 y^{2} - 126 x + 2 y = - 203$

Step 1.2

Complete the square for $18 x^{2} - 126 x$ .

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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 $c$ .

$a = 18$

$b = - 126$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 126}{2 \cdot 18}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 126$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 126$ .

$d = \frac{2 \cdot - 63}{2 \cdot 18}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 18$ .

$d = \frac{2 \cdot - 63}{2 (18)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 63}{2 \cdot 18}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 63}{18}$

Step 1.2.3.2.2

Cancel the common factor of $- 63$ and $18$ .

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Step 1.2.3.2.2.1

Factor $9$ out of $- 63$ .

$d = \frac{9 (- 7)}{18}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $9$ out of $18$ .

$d = \frac{9 \cdot - 7}{9 \cdot 2}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 7}{9 \cdot 2}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 7}{2}$

Step 1.2.3.2.3

Move the negative in front of the fraction.

$d = - \frac{7}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 126)}^{2}}{4 \cdot 18}$

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 $- 126$ to the power of $2$ .

$e = 0 - \frac{15876}{4 \cdot 18}$

Step 1.2.4.2.1.2

Multiply $4$ by $18$ .

$e = 0 - \frac{15876}{72}$

Step 1.2.4.2.1.3

Cancel the common factor of $15876$ and $72$ .

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Step 1.2.4.2.1.3.1

Factor $36$ out of $15876$ .

$e = 0 - \frac{36 (441)}{72}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $36$ out of $72$ .

$e = 0 - \frac{36 \cdot 441}{36 \cdot 2}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{36 \cdot 441}{36 \cdot 2}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{441}{2}$

Step 1.2.4.2.2

Subtract $\frac{441}{2}$ from $0$ .

$e = - \frac{441}{2}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $18 {(x - \frac{7}{2})}^{2} - \frac{441}{2}$ .

$18 {(x - \frac{7}{2})}^{2} - \frac{441}{2}$

Step 1.3

Substitute $18 {(x - \frac{7}{2})}^{2} - \frac{441}{2}$ for $18 x^{2} - 126 x$ in the equation $18 x^{2} + 2 y^{2} - 126 x + 2 y = - 203$ .

$18 {(x - \frac{7}{2})}^{2} - \frac{441}{2} + 2 y^{2} + 2 y = - 203$

Step 1.4

Move $- \frac{441}{2}$ to the right side of the equation by adding $\frac{441}{2}$ to both sides.

$18 {(x - \frac{7}{2})}^{2} + 2 y^{2} + 2 y = - 203 + \frac{441}{2}$

Step 1.5

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

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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 $c$ .

$a = 2$

$b = 2$

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

Step 1.5.3.2

Cancel the common factor of $2$ .

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Step 1.5.3.2.1

Cancel the common factor.

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

Step 1.5.3.2.2

Rewrite the expression.

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

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 $e = c - \frac{b^{2}}{4 a}$ .

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

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

Cancel the common factor of $2^{2}$ and $2$ .

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Step 1.5.4.2.1.1.1

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

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

Step 1.5.4.2.1.1.2

Cancel the common factors.

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Step 1.5.4.2.1.1.2.1

Factor $2$ out of $4 \cdot 2$ .

$e = 0 - \frac{2 \cdot 2}{2 (2 \cdot 2)}$

Step 1.5.4.2.1.1.2.2

Cancel the common factor.

$e = 0 - \frac{2 \cdot 2}{2 (2 \cdot 2)}$

Step 1.5.4.2.1.1.2.3

Rewrite the expression.

$e = 0 - \frac{2}{2 \cdot 2}$

Step 1.5.4.2.1.2

Cancel the common factor of $2$ .

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Step 1.5.4.2.1.2.1

Cancel the common factor.

$e = 0 - \frac{2}{2 \cdot 2}$

Step 1.5.4.2.1.2.2

Rewrite the expression.

$e = 0 - \frac{1}{2}$

Step 1.5.4.2.2

Subtract $\frac{1}{2}$ from $0$ .

$e = - \frac{1}{2}$

Step 1.5.5

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

$2 {(y + \frac{1}{2})}^{2} - \frac{1}{2}$

Step 1.6

Substitute $2 {(y + \frac{1}{2})}^{2} - \frac{1}{2}$ for $2 y^{2} + 2 y$ in the equation $18 x^{2} + 2 y^{2} - 126 x + 2 y = - 203$ .

$18 {(x - \frac{7}{2})}^{2} + 2 {(y + \frac{1}{2})}^{2} - \frac{1}{2} = - 203 + \frac{441}{2}$

Step 1.7

Move $- \frac{1}{2}$ to the right side of the equation by adding $\frac{1}{2}$ to both sides.

$18 {(x - \frac{7}{2})}^{2} + 2 {(y + \frac{1}{2})}^{2} = - 203 + \frac{441}{2} + \frac{1}{2}$

Step 1.8

Simplify $- 203 + \frac{441}{2} + \frac{1}{2}$ .

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Step 1.8.1

Combine the numerators over the common denominator.

$18 {(x - \frac{7}{2})}^{2} + 2 {(y + \frac{1}{2})}^{2} = - 203 + \frac{441 + 1}{2}$

Step 1.8.2

Simplify the expression.

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Step 1.8.2.1

Add $441$ and $1$ .

$18 {(x - \frac{7}{2})}^{2} + 2 {(y + \frac{1}{2})}^{2} = - 203 + \frac{442}{2}$

Step 1.8.2.2

Divide $442$ by $2$ .

$18 {(x - \frac{7}{2})}^{2} + 2 {(y + \frac{1}{2})}^{2} = - 203 + 221$

Step 1.8.2.3

Add $- 203$ and $221$ .

$18 {(x - \frac{7}{2})}^{2} + 2 {(y + \frac{1}{2})}^{2} = 18$

Step 1.9

Divide each term by $18$ to make the right side equal to one.

$\frac{18 {(x - \frac{7}{2})}^{2}}{18} + \frac{2 {(y + \frac{1}{2})}^{2}}{18} = \frac{18}{18}$

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 $1$ .

${(x - \frac{7}{2})}^{2} + \frac{{(y + \frac{1}{2})}^{2}}{9} = 1$

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.

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

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.

$a = 3$

$b = 1$

$k = - \frac{1}{2}$

$h = \frac{7}{2}$

Step 4

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

$(\frac{7}{2}, - \frac{1}{2})$

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.

$\sqrt{{(3)}^{2} - {(1)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $3$ to the power of $2$ .

$\sqrt{9 - {(1)}^{2}}$

Step 5.3.2

One to any power is one.

$\sqrt{9 - 1 \cdot 1}$

Step 5.3.3

Multiply $- 1$ by $1$ .

$\sqrt{9 - 1}$

Step 5.3.4

Subtract $1$ from $9$ .

$\sqrt{8}$

Step 5.3.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.3.5.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 5.3.5.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 2}$

Step 5.3.6

Pull terms out from under the radical.

$2 \sqrt{2}$

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 $k$ .

$(h, k + a)$

Step 6.2

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

$(\frac{7}{2}, - \frac{1}{2} + 3)$

Step 6.3

Simplify.

$(\frac{7}{2}, \frac{5}{2})$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.5

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

$(\frac{7}{2}, - \frac{1}{2} - (3))$

Step 6.6

Simplify.

$(\frac{7}{2}, - \frac{7}{2})$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(\frac{7}{2}, \frac{5}{2})$

${Vertex}_{2}$ : $(\frac{7}{2}, - \frac{7}{2})$

${Vertex}_{1}$ : $(\frac{7}{2}, \frac{5}{2})$

${Vertex}_{2}$ : $(\frac{7}{2}, - \frac{7}{2})$

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 $k$ .

$(h, k + c)$

Step 7.2

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

$(\frac{7}{2}, - \frac{1}{2} + 2 \sqrt{2})$

Step 7.3

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.4

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

$(\frac{7}{2}, - \frac{1}{2} - (2 \sqrt{2}))$

Step 7.5

Simplify.

$(\frac{7}{2}, - \frac{1}{2} - 2 \sqrt{2})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(\frac{7}{2}, - \frac{1}{2} + 2 \sqrt{2})$

${Focus}_{2}$ : $(\frac{7}{2}, - \frac{1}{2} - 2 \sqrt{2})$

${Focus}_{1}$ : $(\frac{7}{2}, - \frac{1}{2} + 2 \sqrt{2})$

${Focus}_{2}$ : $(\frac{7}{2}, - \frac{1}{2} - 2 \sqrt{2})$

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.

$\frac{\sqrt{{(3)}^{2} - {(1)}^{2}}}{3}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 - 1^{2}}}{3}$

Step 8.3.2

One to any power is one.

$\frac{\sqrt{9 - 1 \cdot 1}}{3}$

Step 8.3.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{9 - 1}}{3}$

Step 8.3.4

Subtract $1$ from $9$ .

$\frac{\sqrt{8}}{3}$

Step 8.3.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 8.3.5.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)}}{3}$

Step 8.3.5.2

Rewrite $4$ as $2^{2}$ .

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

Step 8.3.6

Pull terms out from under the radical.

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

Step 9

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

Center: $(\frac{7}{2}, - \frac{1}{2})$

${Vertex}_{1}$ : $(\frac{7}{2}, \frac{5}{2})$

${Vertex}_{2}$ : $(\frac{7}{2}, - \frac{7}{2})$

${Focus}_{1}$ : $(\frac{7}{2}, - \frac{1}{2} + 2 \sqrt{2})$

${Focus}_{2}$ : $(\frac{7}{2}, - \frac{1}{2} - 2 \sqrt{2})$

Eccentricity: $\frac{2 \sqrt{2}}{3}$

Step 10