Pre-Algebra Examples

$4 x^{2} + y^{2} - 32 x - 12 y + 84 = 0$

Step 1

Find the standard form of the ellipse.

Tap for more steps...

Step 1.1

Subtract $84$ from both sides of the equation.

$4 x^{2} + y^{2} - 32 x - 12 y = - 84$

Step 1.2

Complete the square for $4 x^{2} - 32 x$ .

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 $c$ .

$a = 4$

$b = - 32$

$c = 0$

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

$d = \frac{- 32}{2 \cdot 4}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 32$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $- 32$ .

$d = \frac{2 \cdot - 16}{2 \cdot 4}$

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 $2 \cdot 4$ .

$d = \frac{2 \cdot - 16}{2 (4)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 16}{2 \cdot 4}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 16}{4}$

Step 1.2.3.2.2

Cancel the common factor of $- 16$ and $4$ .

Tap for more steps...

Step 1.2.3.2.2.1

Factor $4$ out of $- 16$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.2.2.1

Factor $4$ out of $4$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = 0 - \frac{{(- 32)}^{2}}{4 \cdot 4}$

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 $- 32$ to the power of $2$ .

$e = 0 - \frac{1024}{4 \cdot 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{1024}{16}$

Step 1.2.4.2.1.3

Divide $1024$ by $16$ .

$e = 0 - 1 \cdot 64$

Step 1.2.4.2.1.4

Multiply $- 1$ by $64$ .

$e = 0 - 64$

Step 1.2.4.2.2

Subtract $64$ from $0$ .

$e = - 64$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 4)}^{2} - 64$ .

$4 {(x - 4)}^{2} - 64$

Step 1.3

Substitute $4 {(x - 4)}^{2} - 64$ for $4 x^{2} - 32 x$ in the equation $4 x^{2} + y^{2} - 32 x - 12 y = - 84$ .

$4 {(x - 4)}^{2} - 64 + y^{2} - 12 y = - 84$

Step 1.4

Move $- 64$ to the right side of the equation by adding $64$ to both sides.

$4 {(x - 4)}^{2} + y^{2} - 12 y = - 84 + 64$

Step 1.5

Complete the square for $y^{2} - 12 y$ .

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 $c$ .

$a = 1$

$b = - 12$

$c = 0$

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

$d = \frac{- 12}{2 \cdot 1}$

Step 1.5.3.2

Cancel the common factor of $- 12$ and $2$ .

Tap for more steps...

Step 1.5.3.2.1

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 1}$

Step 1.5.3.2.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 6}{2 (1)}$

Step 1.5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 1}$

Step 1.5.3.2.2.3

Rewrite the expression.

$d = \frac{- 6}{1}$

Step 1.5.3.2.2.4

Divide $- 6$ by $1$ .

$d = - 6$

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

$e = 0 - \frac{{(- 12)}^{2}}{4 \cdot 1}$

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 $- 12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 1}$

Step 1.5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{144}{4}$

Step 1.5.4.2.1.3

Divide $144$ by $4$ .

$e = 0 - 1 \cdot 36$

Step 1.5.4.2.1.4

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Step 1.5.4.2.2

Subtract $36$ from $0$ .

$e = - 36$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 6)}^{2} - 36$ .

${(y - 6)}^{2} - 36$

Step 1.6

Substitute ${(y - 6)}^{2} - 36$ for $y^{2} - 12 y$ in the equation $4 x^{2} + y^{2} - 32 x - 12 y = - 84$ .

$4 {(x - 4)}^{2} + {(y - 6)}^{2} - 36 = - 84 + 64$

Step 1.7

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

$4 {(x - 4)}^{2} + {(y - 6)}^{2} = - 84 + 64 + 36$

Step 1.8

Simplify $- 84 + 64 + 36$ .

Tap for more steps...

Step 1.8.1

Add $- 84$ and $64$ .

$4 {(x - 4)}^{2} + {(y - 6)}^{2} = - 20 + 36$

Step 1.8.2

Add $- 20$ and $36$ .

$4 {(x - 4)}^{2} + {(y - 6)}^{2} = 16$

Step 1.9

Divide each term by $16$ to make the right side equal to one.

$\frac{4 {(x - 4)}^{2}}{16} + \frac{{(y - 6)}^{2}}{16} = \frac{16}{16}$

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 $1$ .

$\frac{{(x - 4)}^{2}}{4} + \frac{{(y - 6)}^{2}}{16} = 1$

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.

$a = 4$

$b = 2$

$k = 6$

$h = 4$

Step 4

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

$(4, 6)$

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.

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

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

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

$\sqrt{16 - {(2)}^{2}}$

Step 5.3.2

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

$\sqrt{16 - 1 \cdot 4}$

Step 5.3.3

Multiply $- 1$ by $4$ .

$\sqrt{16 - 4}$

Step 5.3.4

Subtract $4$ from $16$ .

$\sqrt{12}$

Step 5.3.5

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 5.3.5.1

Factor $4$ out of $12$ .

$\sqrt{4 (3)}$

Step 5.3.5.2

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

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

Step 5.3.6

Pull terms out from under the radical.

$2 \sqrt{3}$

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.

$(4, 6 + 4)$

Step 6.3

Simplify.

$(4, 10)$

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.

$(4, 6 - (4))$

Step 6.6

Simplify.

$(4, 2)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(4, 10)$

${Vertex}_{2}$ : $(4, 2)$

${Vertex}_{1}$ : $(4, 10)$

${Vertex}_{2}$ : $(4, 2)$

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.

$(4, 6 + 2 \sqrt{3})$

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.

$(4, 6 - (2 \sqrt{3}))$

Step 7.5

Simplify.

$(4, 6 - 2 \sqrt{3})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(4, 6 + 2 \sqrt{3})$

${Focus}_{2}$ : $(4, 6 - 2 \sqrt{3})$

${Focus}_{1}$ : $(4, 6 + 2 \sqrt{3})$

${Focus}_{2}$ : $(4, 6 - 2 \sqrt{3})$

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.

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

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 $4$ to the power of $2$ .

$\frac{\sqrt{16 - 2^{2}}}{4}$

Step 8.3.1.2

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

$\frac{\sqrt{16 - 1 \cdot 4}}{4}$

Step 8.3.1.3

Multiply $- 1$ by $4$ .

$\frac{\sqrt{16 - 4}}{4}$

Step 8.3.1.4

Subtract $4$ from $16$ .

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

Step 8.3.1.5

Rewrite $12$ as $2^{2} \cdot 3$ .

Tap for more steps...

Step 8.3.1.5.1

Factor $4$ out of $12$ .

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

Step 8.3.1.5.2

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

$\frac{\sqrt{2^{2} \cdot 3}}{4}$

Step 8.3.1.6

Pull terms out from under the radical.

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

Step 8.3.2

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 8.3.2.1

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 8.3.2.2

Cancel the common factors.

Tap for more steps...

Step 8.3.2.2.1

Factor $2$ out of $4$ .

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

Step 8.3.2.2.2

Cancel the common factor.

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

Step 8.3.2.2.3

Rewrite the expression.

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

Step 9

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

Center: $(4, 6)$

${Vertex}_{1}$ : $(4, 10)$

${Vertex}_{2}$ : $(4, 2)$

${Focus}_{1}$ : $(4, 6 + 2 \sqrt{3})$

${Focus}_{2}$ : $(4, 6 - 2 \sqrt{3})$

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

Step 10