Precalculus Examples

$x^{2} + 10 x + 41 + y^{2} - 8 y = 36$

Step 1

Move all terms not containing a variable to the right side of the equation.

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Step 1.1

Subtract $41$ from both sides of the equation.

$x^{2} + 10 x + y^{2} - 8 y = 36 - 41$

Step 1.2

Subtract $41$ from $36$ .

$x^{2} + 10 x + y^{2} - 8 y = - 5$

Step 2

Complete the square for $x^{2} + 10 x$ .

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Step 2.1

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

$a = 1$

$b = 10$

$c = 0$

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

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

Step 2.3.2

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

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Step 2.3.2.1

Factor $2$ out of $10$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $5$ by $1$ .

$d = 5$

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

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

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 $10$ to the power of $2$ .

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Step 2.4.2.1.4

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Step 2.4.2.2

Subtract $25$ from $0$ .

$e = - 25$

Step 2.5

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

${(x + 5)}^{2} - 25$

Step 3

Substitute ${(x + 5)}^{2} - 25$ for $x^{2} + 10 x$ in the equation $x^{2} + 10 x + y^{2} - 8 y = - 5$ .

${(x + 5)}^{2} - 25 + y^{2} - 8 y = - 5$

Step 4

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

${(x + 5)}^{2} + y^{2} - 8 y = - 5 + 25$

Step 5

Complete the square for $y^{2} - 8 y$ .

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Step 5.1

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

$a = 1$

$b = - 8$

$c = 0$

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

$d = \frac{- 8}{2 \cdot 1}$

Step 5.3.2

Cancel the common factor of $- 8$ and $2$ .

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Step 5.3.2.1

Factor $2$ out of $- 8$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

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

$d = \frac{2 \cdot - 4}{2 (1)}$

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

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 $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{64}{4}$

Step 5.4.2.1.3

Divide $64$ by $4$ .

$e = 0 - 1 \cdot 16$

Step 5.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 5.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 4)}^{2} - 16$ .

${(y - 4)}^{2} - 16$

Step 6

Substitute ${(y - 4)}^{2} - 16$ for $y^{2} - 8 y$ in the equation $x^{2} + 10 x + y^{2} - 8 y = - 5$ .

${(x + 5)}^{2} + {(y - 4)}^{2} - 16 = - 5 + 25$

Step 7

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

${(x + 5)}^{2} + {(y - 4)}^{2} = - 5 + 25 + 16$

Step 8

Simplify $- 5 + 25 + 16$ .

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Step 8.1

Add $- 5$ and $25$ .

${(x + 5)}^{2} + {(y - 4)}^{2} = 20 + 16$

Step 8.2

Add $20$ and $16$ .

${(x + 5)}^{2} + {(y - 4)}^{2} = 36$

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.

$r = 6$

$h = - 5$

$k = 4$

Step 11

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

Center: $(- 5, 4)$

Step 12

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

Center: $(- 5, 4)$

Radius: $6$

Step 13