Pre-Algebra Examples

$9 x^{2} + 4 y^{2} - 54 x - 8 y - 59 = 0$

Step 1

Find the standard form of the ellipse.

Tap for more steps...

Step 1.1

Add $59$ to both sides of the equation.

$9 x^{2} + 4 y^{2} - 54 x - 8 y = 59$

Step 1.2

Complete the square for $9 x^{2} - 54 x$ .

Tap for more steps...

Step 1.2.1

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

$a = 9$

$b = - 54$

$c = 0$

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

Tap for more steps...

Step 1.2.3.1

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

$d = \frac{- 54}{2 \cdot 9}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 54$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $- 54$ .

$d = \frac{2 \cdot - 27}{2 \cdot 9}$

Step 1.2.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 27}{2 (9)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 27}{2 \cdot 9}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 27}{9}$

Step 1.2.3.2.2

Cancel the common factor of $- 27$ and $9$ .

Tap for more steps...

Step 1.2.3.2.2.1

Factor $9$ out of $- 27$ .

$d = \frac{9 \cdot - 3}{9}$

Step 1.2.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 3}{9 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 3}{9 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 3}{1}$

Step 1.2.3.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$e = 0 - \frac{{(- 54)}^{2}}{4 \cdot 9}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Raise $- 54$ to the power of $2$ .

$e = 0 - \frac{2916}{4 \cdot 9}$

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{2916}{36}$

Step 1.2.4.2.1.3

Divide $2916$ by $36$ .

$e = 0 - 1 \cdot 81$

Step 1.2.4.2.1.4

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Step 1.2.4.2.2

Subtract $81$ from $0$ .

$e = - 81$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 3)}^{2} - 81$ .

$9 {(x - 3)}^{2} - 81$

Step 1.3

Substitute $9 {(x - 3)}^{2} - 81$ for $9 x^{2} - 54 x$ in the equation $9 x^{2} + 4 y^{2} - 54 x - 8 y = 59$ .

$9 {(x - 3)}^{2} - 81 + 4 y^{2} - 8 y = 59$

Step 1.4

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

$9 {(x - 3)}^{2} + 4 y^{2} - 8 y = 59 + 81$

Step 1.5

Complete the square for $4 y^{2} - 8 y$ .

Tap for more steps...

Step 1.5.1

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

$a = 4$

$b = - 8$

$c = 0$

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

Tap for more steps...

Step 1.5.3.1

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

$d = \frac{- 8}{2 \cdot 4}$

Step 1.5.3.2

Simplify the right side.

Tap for more steps...

Step 1.5.3.2.1

Cancel the common factor of $- 8$ and $2$ .

Tap for more steps...

Step 1.5.3.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 4}$

Step 1.5.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.1.2.1

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

$d = \frac{2 \cdot - 4}{2 (4)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 4}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 4}{4}$

Step 1.5.3.2.2

Cancel the common factor of $- 4$ and $4$ .

Tap for more steps...

Step 1.5.3.2.2.1

Factor $4$ out of $- 4$ .

$d = \frac{4 \cdot - 1}{4}$

Step 1.5.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 1}{4 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot - 1}{4 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

Step 1.5.4

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

Tap for more steps...

Step 1.5.4.1

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 4}$

Step 1.5.4.2

Simplify the right side.

Tap for more steps...

Step 1.5.4.2.1

Simplify each term.

Tap for more steps...

Step 1.5.4.2.1.1

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Step 1.5.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Step 1.5.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 1.5.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.5.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 1)}^{2} - 4$ .

$4 {(y - 1)}^{2} - 4$

Step 1.6

Substitute $4 {(y - 1)}^{2} - 4$ for $4 y^{2} - 8 y$ in the equation $9 x^{2} + 4 y^{2} - 54 x - 8 y = 59$ .

$9 {(x - 3)}^{2} + 4 {(y - 1)}^{2} - 4 = 59 + 81$

Step 1.7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

$9 {(x - 3)}^{2} + 4 {(y - 1)}^{2} = 59 + 81 + 4$

Step 1.8

Simplify $59 + 81 + 4$ .

Tap for more steps...

Step 1.8.1

Add $59$ and $81$ .

$9 {(x - 3)}^{2} + 4 {(y - 1)}^{2} = 140 + 4$

Step 1.8.2

Add $140$ and $4$ .

$9 {(x - 3)}^{2} + 4 {(y - 1)}^{2} = 144$

Step 1.9

Divide each term by $144$ to make the right side equal to one.

$\frac{9 {(x - 3)}^{2}}{144} + \frac{4 {(y - 1)}^{2}}{144} = \frac{144}{144}$

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 $1$ .

$\frac{{(x - 3)}^{2}}{16} + \frac{{(y - 1)}^{2}}{36} = 1$

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.

$a = 6$

$b = 4$

$k = 1$

$h = 3$

Step 4

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

$(3, 1)$

Step 5

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

Tap for more steps...

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.

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

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

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

$\sqrt{36 - {(4)}^{2}}$

Step 5.3.2

Raise $4$ to the power of $2$ .

$\sqrt{36 - 1 \cdot 16}$

Step 5.3.3

Multiply $- 1$ by $16$ .

$\sqrt{36 - 16}$

Step 5.3.4

Subtract $16$ from $36$ .

$\sqrt{20}$

Step 5.3.5

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 5.3.5.1

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Step 5.3.5.2

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

$\sqrt{2^{2} \cdot 5}$

Step 5.3.6

Pull terms out from under the radical.

$2 \sqrt{5}$

Step 6

Find the vertices.

Tap for more steps...

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.

$(3, 1 + 6)$

Step 6.3

Simplify.

$(3, 7)$

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.

$(3, 1 - (6))$

Step 6.6

Simplify.

$(3, - 5)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(3, 7)$

${Vertex}_{2}$ : $(3, - 5)$

${Vertex}_{1}$ : $(3, 7)$

${Vertex}_{2}$ : $(3, - 5)$

Step 7

Find the foci.

Tap for more steps...

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.

$(3, 1 + 2 \sqrt{5})$

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.

$(3, 1 - (2 \sqrt{5}))$

Step 7.5

Simplify.

$(3, 1 - 2 \sqrt{5})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(3, 1 + 2 \sqrt{5})$

${Focus}_{2}$ : $(3, 1 - 2 \sqrt{5})$

${Focus}_{1}$ : $(3, 1 + 2 \sqrt{5})$

${Focus}_{2}$ : $(3, 1 - 2 \sqrt{5})$

Step 8

Find the eccentricity.

Tap for more steps...

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.

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

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Simplify the numerator.

Tap for more steps...

Step 8.3.1.1

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

$\frac{\sqrt{36 - 4^{2}}}{6}$

Step 8.3.1.2

Raise $4$ to the power of $2$ .

$\frac{\sqrt{36 - 1 \cdot 16}}{6}$

Step 8.3.1.3

Multiply $- 1$ by $16$ .

$\frac{\sqrt{36 - 16}}{6}$

Step 8.3.1.4

Subtract $16$ from $36$ .

$\frac{\sqrt{20}}{6}$

Step 8.3.1.5

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 8.3.1.5.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{6}$

Step 8.3.1.5.2

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

$\frac{\sqrt{2^{2} \cdot 5}}{6}$

Step 8.3.1.6

Pull terms out from under the radical.

$\frac{2 \sqrt{5}}{6}$

Step 8.3.2

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

Tap for more steps...

Step 8.3.2.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 (\sqrt{5})}{6}$

Step 8.3.2.2

Cancel the common factors.

Tap for more steps...

Step 8.3.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \sqrt{5}}{2 \cdot 3}$

Step 8.3.2.2.2

Cancel the common factor.

$\frac{2 \sqrt{5}}{2 \cdot 3}$

Step 8.3.2.2.3

Rewrite the expression.

$\frac{\sqrt{5}}{3}$

Step 9

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

Center: $(3, 1)$

${Vertex}_{1}$ : $(3, 7)$

${Vertex}_{2}$ : $(3, - 5)$

${Focus}_{1}$ : $(3, 1 + 2 \sqrt{5})$

${Focus}_{2}$ : $(3, 1 - 2 \sqrt{5})$

Eccentricity: $\frac{\sqrt{5}}{3}$

Step 10