Pre-Algebra Examples

$4 x^{2} + 4 x + 4 y^{2} - 24 y - 219 = 0$

Step 1

Add $219$ to both sides of the equation.

$4 x^{2} + 4 x + 4 y^{2} - 24 y = 219$

Step 2

Divide both sides of the equation by $4$ .

$x^{2} + x + y^{2} - 6 y = \frac{219}{4}$

Step 3

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

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Step 3.1

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

$a = 1$

$b = 1$

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $1$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

One to any power is one.

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

Step 3.4.2.1.2

Multiply $4$ by $1$ .

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

Step 3.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{2})}^{2} - \frac{1}{4}$ .

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

Step 4

Substitute ${(x + \frac{1}{2})}^{2} - \frac{1}{4}$ for $x^{2} + x$ in the equation $x^{2} + x + y^{2} - 6 y = \frac{219}{4}$ .

${(x + \frac{1}{2})}^{2} - \frac{1}{4} + y^{2} - 6 y = \frac{219}{4}$

Step 5

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x + \frac{1}{2})}^{2} + y^{2} - 6 y = \frac{219}{4} + \frac{1}{4}$

Step 6

Complete the square for $y^{2} - 6 y$ .

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Step 6.1

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

$a = 1$

$b = - 6$

$c = 0$

Step 6.2

Consider the vertex form of a parabola.

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

Step 6.3

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

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Step 6.3.1

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

$d = \frac{- 6}{2 \cdot 1}$

Step 6.3.2

Cancel the common factor of $- 6$ and $2$ .

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Step 6.3.2.1

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

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

$d = \frac{2 \cdot - 3}{2 (1)}$

Step 6.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Step 6.3.2.2.3

Rewrite the expression.

$d = \frac{- 3}{1}$

Step 6.3.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

Step 6.4

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

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Step 6.4.1

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

$e = 0 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Raise $- 6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot 1}$

Step 6.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{36}{4}$

Step 6.4.2.1.3

Divide $36$ by $4$ .

$e = 0 - 1 \cdot 9$

Step 6.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Step 6.4.2.2

Subtract $9$ from $0$ .

$e = - 9$

Step 6.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 3)}^{2} - 9$ .

${(y - 3)}^{2} - 9$

Step 7

Substitute ${(y - 3)}^{2} - 9$ for $y^{2} - 6 y$ in the equation $x^{2} + x + y^{2} - 6 y = \frac{219}{4}$ .

${(x + \frac{1}{2})}^{2} + {(y - 3)}^{2} - 9 = \frac{219}{4} + \frac{1}{4}$

Step 8

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

${(x + \frac{1}{2})}^{2} + {(y - 3)}^{2} = \frac{219}{4} + \frac{1}{4} + 9$

Step 9

Simplify $\frac{219}{4} + \frac{1}{4} + 9$ .

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Step 9.1

Combine the numerators over the common denominator.

${(x + \frac{1}{2})}^{2} + {(y - 3)}^{2} = 9 + \frac{219 + 1}{4}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Add $219$ and $1$ .

${(x + \frac{1}{2})}^{2} + {(y - 3)}^{2} = 9 + \frac{220}{4}$

Step 9.2.2

Divide $220$ by $4$ .

${(x + \frac{1}{2})}^{2} + {(y - 3)}^{2} = 9 + 55$

Step 9.2.3

Add $9$ and $55$ .

${(x + \frac{1}{2})}^{2} + {(y - 3)}^{2} = 64$

Step 10

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 11

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.

$r = 8$

$h = - \frac{1}{2}$

$k = 3$

Step 12

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

Center: $(- \frac{1}{2}, 3)$

Step 13

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

Center: $(- \frac{1}{2}, 3)$

Radius: $8$

Step 14