Precalculus Examples

$16 x^{2} + y^{2} + 14 y = - 33$

Step 1

Complete the square for $y^{2} + 14 y$ .

Tap for more steps...

Step 1.1

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

$a = 1$

$b = 14$

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

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

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$d = \frac{14}{2 \cdot 1}$

Step 1.3.2

Cancel the common factor of $14$ and $2$ .

Tap for more steps...

Step 1.3.2.1

Factor $2$ out of $14$ .

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Step 1.3.2.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.2.1

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

$d = \frac{2 \cdot 7}{2 (1)}$

Step 1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Step 1.3.2.2.3

Rewrite the expression.

$d = \frac{7}{1}$

Step 1.3.2.2.4

Divide $7$ by $1$ .

$d = 7$

Step 1.4

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

Tap for more steps...

Step 1.4.1

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

$e = 0 - \frac{14^{2}}{4 \cdot 1}$

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.1.1

Raise $14$ to the power of $2$ .

$e = 0 - \frac{196}{4 \cdot 1}$

Step 1.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{196}{4}$

Step 1.4.2.1.3

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Step 1.4.2.1.4

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Step 1.4.2.2

Subtract $49$ from $0$ .

$e = - 49$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 7)}^{2} - 49$ .

${(y + 7)}^{2} - 49$

Step 2

Substitute ${(y + 7)}^{2} - 49$ for $y^{2} + 14 y$ in the equation $16 x^{2} + y^{2} + 14 y = - 33$ .

$16 x^{2} + {(y + 7)}^{2} - 49 = - 33$

Step 3

Move $- 49$ to the right side of the equation by adding $49$ to both sides.

$16 x^{2} + {(y + 7)}^{2} = - 33 + 49$

Step 4

Add $- 33$ and $49$ .

$16 x^{2} + {(y + 7)}^{2} = 16$

Step 5

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

$\frac{16 x^{2}}{16} + \frac{{(y + 7)}^{2}}{16} = \frac{16}{16}$

Step 6

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 $1$ .

$x^{2} + \frac{{(y + 7)}^{2}}{16} = 1$