Trigonometry Examples

$x^{2} + y^{2} + 22 x + 14 y - 55 = 0$

Step 1

Add $55$ to both sides of the equation.

$x^{2} + y^{2} + 22 x + 14 y = 55$

Step 2

Complete the square for $x^{2} + 22 x$ .

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Step 2.1

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

$a = 1$

$b = 22$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

$d = \frac{22}{2 \cdot 1}$

Step 2.3.2

Cancel the common factor of $22$ and $2$ .

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Step 2.3.2.1

Factor $2$ out of $22$ .

$d = \frac{2 \cdot 11}{2 \cdot 1}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

$d = \frac{2 \cdot 11}{2 (1)}$

Step 2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 11}{2 \cdot 1}$

Step 2.3.2.2.3

Rewrite the expression.

$d = \frac{11}{1}$

Step 2.3.2.2.4

Divide $11$ by $1$ .

$d = 11$

Step 2.4

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

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Step 2.4.1

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

$e = 0 - \frac{22^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $22$ to the power of $2$ .

$e = 0 - \frac{484}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{484}{4}$

Step 2.4.2.1.3

Divide $484$ by $4$ .

$e = 0 - 1 \cdot 121$

Step 2.4.2.1.4

Multiply $- 1$ by $121$ .

$e = 0 - 121$

Step 2.4.2.2

Subtract $121$ from $0$ .

$e = - 121$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 11)}^{2} - 121$ .

${(x + 11)}^{2} - 121$

Step 3

Substitute ${(x + 11)}^{2} - 121$ for $x^{2} + 22 x$ in the equation $x^{2} + y^{2} + 22 x + 14 y = 55$ .

${(x + 11)}^{2} - 121 + y^{2} + 14 y = 55$

Step 4

Move $- 121$ to the right side of the equation by adding $121$ to both sides.

${(x + 11)}^{2} + y^{2} + 14 y = 55 + 121$

Step 5

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

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Step 5.1

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 5.2

Consider the vertex form of a parabola.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

Factor $2$ out of $14$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

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

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.4

Divide $7$ by $1$ .

$d = 7$

Step 5.4

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

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Step 5.4.1

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 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

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

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

Step 5.4.2.1.2

Multiply $4$ by $1$ .

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

Step 5.4.2.1.3

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Step 5.4.2.1.4

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Step 5.4.2.2

Subtract $49$ from $0$ .

$e = - 49$

Step 5.5

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

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

Step 6

Substitute ${(y + 7)}^{2} - 49$ for $y^{2} + 14 y$ in the equation $x^{2} + y^{2} + 22 x + 14 y = 55$ .

${(x + 11)}^{2} + {(y + 7)}^{2} - 49 = 55 + 121$

Step 7

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

${(x + 11)}^{2} + {(y + 7)}^{2} = 55 + 121 + 49$

Step 8

Simplify $55 + 121 + 49$ .

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Step 8.1

Add $55$ and $121$ .

${(x + 11)}^{2} + {(y + 7)}^{2} = 176 + 49$

Step 8.2

Add $176$ and $49$ .

${(x + 11)}^{2} + {(y + 7)}^{2} = 225$