Precalculus Examples

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

Step 1

Find the standard form of the ellipse.

Tap for more steps...

Step 1.1

Subtract $4$ from both sides of the equation.

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

Step 1.2

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

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 $c$ .

$a = 1$

$b = 4$

$c = 0$

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

$d = \frac{4}{2 \cdot 1}$

Step 1.2.3.2

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

Tap for more steps...

Step 1.2.3.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.2.1

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

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

Step 1.2.3.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

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

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

Cancel the common factor of $4^{2}$ and $4$ .

Tap for more steps...

Step 1.2.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

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

Step 1.2.4.2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 1$ .

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

Step 1.2.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.1.1.2.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 1.2.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.2.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.2.5

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

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

Step 1.3

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

${(x + 2)}^{2} - 4 + 4 y^{2} - 8 y = - 4$

Step 1.4

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

${(x + 2)}^{2} + 4 y^{2} - 8 y = - 4 + 4$

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 $x^{2} + 4 x + 4 y^{2} - 8 y = - 4$ .

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

Step 1.7

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

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

Step 1.8

Simplify $- 4 + 4 + 4$ .

Tap for more steps...

Step 1.8.1

Add $- 4$ and $4$ .

${(x + 2)}^{2} + 4 {(y - 1)}^{2} = 0 + 4$

Step 1.8.2

Add $0$ and $4$ .

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

Step 1.9

Divide each term by $4$ to make the right side equal to one.

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

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 $1$ .

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

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.

$a = 2$

$b = 1$

$k = 1$

$h = - 2$

Step 4

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

$(- 2, 1)$

Step 5