Algebra Examples

$x^{2} + y^{2} + 14 x - 10 y + 65 = 0$

Step 1

Subtract $65$ from both sides of the equation.

$x^{2} + y^{2} + 14 x - 10 y = - 65$

Step 2

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

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Step 2.1

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

$a = 1$

$b = 14$

$c = 0$

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

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

Step 2.3.2

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

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Step 2.3.2.1

Factor $2$ out of $14$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $7$ by $1$ .

$d = 7$

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

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

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 $14$ to the power of $2$ .

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Step 2.4.2.1.4

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Step 2.4.2.2

Subtract $49$ from $0$ .

$e = - 49$

Step 2.5

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

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

Step 3

Substitute ${(x + 7)}^{2} - 49$ for $x^{2} + 14 x$ in the equation $x^{2} + y^{2} + 14 x - 10 y = - 65$ .

${(x + 7)}^{2} - 49 + y^{2} - 10 y = - 65$

Step 4

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

${(x + 7)}^{2} + y^{2} - 10 y = - 65 + 49$

Step 5

Complete the square for $y^{2} - 10 y$ .

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Step 5.1

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

$a = 1$

$b = - 10$

$c = 0$

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

$d = \frac{- 10}{2 \cdot 1}$

Step 5.3.2

Cancel the common factor of $- 10$ and $2$ .

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Step 5.3.2.1

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

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

$d = \frac{2 \cdot - 5}{2 (1)}$

Step 5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Step 5.3.2.2.3

Rewrite the expression.

$d = \frac{- 5}{1}$

Step 5.3.2.2.4

Divide $- 5$ by $1$ .

$d = - 5$

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

$e = 0 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

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 $- 10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Step 5.4.2.1.3

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Step 5.4.2.1.4

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Step 5.4.2.2

Subtract $25$ from $0$ .

$e = - 25$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 5)}^{2} - 25$ .

${(y - 5)}^{2} - 25$

Step 6

Substitute ${(y - 5)}^{2} - 25$ for $y^{2} - 10 y$ in the equation $x^{2} + y^{2} + 14 x - 10 y = - 65$ .

${(x + 7)}^{2} + {(y - 5)}^{2} - 25 = - 65 + 49$

Step 7

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

${(x + 7)}^{2} + {(y - 5)}^{2} = - 65 + 49 + 25$

Step 8

Simplify $- 65 + 49 + 25$ .

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Step 8.1

Add $- 65$ and $49$ .

${(x + 7)}^{2} + {(y - 5)}^{2} = - 16 + 25$

Step 8.2

Add $- 16$ and $25$ .

${(x + 7)}^{2} + {(y - 5)}^{2} = 9$

Step 9

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 10

Match the values in this circle to those of the standard form. The variable $r$ represents the radius of the circle, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from origin.

$r = 3$

$h = - 7$

$k = 5$

Step 11

The center of the circle is found at $(h, k)$ .

Center: $(- 7, 5)$

Step 12

These values represent the important values for graphing and analyzing a circle.

Center: $(- 7, 5)$

Radius: $3$

Step 13