Algebra Examples

$144 x^{2} + 144 y^{2} + 72 x + 192 y + 37 = 0$

Step 1

Subtract $37$ from both sides of the equation.

$144 x^{2} + 144 y^{2} + 72 x + 192 y = - 37$

Step 2

Divide both sides of the equation by $144$ .

$x^{2} + y^{2} + \frac{x}{2} + \frac{4 y}{3} = - \frac{37}{144}$

Step 3

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

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Step 3.1

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

$a = 1$

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

$c = 0$

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

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.2.2

Cancel the common factor of $1$ .

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Step 3.3.2.2.1

Cancel the common factor.

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

Step 3.3.2.2.2

Rewrite the expression.

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

Step 3.3.2.3

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 3.3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 3.3.2.3.2

Multiply $2$ by $2$ .

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

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

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

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

Simplify the numerator.

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Step 3.4.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.4.2.1.1.2

One to any power is one.

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

Step 3.4.2.1.1.3

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

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

Step 3.4.2.1.2

Multiply $4$ by $1$ .

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

Step 3.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.4.2.1.4

Multiply $\frac{1}{4} \cdot \frac{1}{4}$ .

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Step 3.4.2.1.4.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

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

Step 3.4.2.1.4.2

Multiply $4$ by $4$ .

$e = 0 - \frac{1}{16}$

Step 3.4.2.2

Subtract $\frac{1}{16}$ from $0$ .

$e = - \frac{1}{16}$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{4})}^{2} - \frac{1}{16}$ .

${(x + \frac{1}{4})}^{2} - \frac{1}{16}$

Step 4

Substitute ${(x + \frac{1}{4})}^{2} - \frac{1}{16}$ for $x^{2} + \frac{x}{2}$ in the equation $x^{2} + y^{2} + \frac{x}{2} + \frac{4 y}{3} = - \frac{37}{144}$ .

${(x + \frac{1}{4})}^{2} - \frac{1}{16} + y^{2} + \frac{4 y}{3} = - \frac{37}{144}$

Step 5

Move $- \frac{1}{16}$ to the right side of the equation by adding $\frac{1}{16}$ to both sides.

${(x + \frac{1}{4})}^{2} + y^{2} + \frac{4 y}{3} = - \frac{37}{144} + \frac{1}{16}$

Step 6

Complete the square for $y^{2} + \frac{4 y}{3}$ .

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Step 6.1

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

$a = 1$

$b = \frac{4}{3}$

$c = 0$

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

$d = \frac{\frac{4}{3}}{2 \cdot 1}$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{4}{3} \cdot \frac{1}{2 \cdot 1}$

Step 6.3.2.2

Cancel the common factor of $2$ .

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Step 6.3.2.2.1

Factor $2$ out of $4$ .

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

Step 6.3.2.2.2

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

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

Step 6.3.2.2.3

Cancel the common factor.

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

Step 6.3.2.2.4

Rewrite the expression.

$d = \frac{2}{3}$

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

$e = 0 - \frac{{(\frac{4}{3})}^{2}}{4 \cdot 1}$

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

Simplify the numerator.

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Step 6.4.2.1.1.1

Apply the product rule to $\frac{4}{3}$ .

$e = 0 - \frac{\frac{4^{2}}{3^{2}}}{4 \cdot 1}$

Step 6.4.2.1.1.2

Raise $4$ to the power of $2$ .

$e = 0 - \frac{\frac{16}{3^{2}}}{4 \cdot 1}$

Step 6.4.2.1.1.3

Raise $3$ to the power of $2$ .

$e = 0 - \frac{\frac{16}{9}}{4 \cdot 1}$

Step 6.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{16}{9}}{4}$

Step 6.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{16}{9} \cdot \frac{1}{4})$

Step 6.4.2.1.4

Cancel the common factor of $4$ .

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Step 6.4.2.1.4.1

Factor $4$ out of $16$ .

$e = 0 - (\frac{4 (4)}{9} \cdot \frac{1}{4})$

Step 6.4.2.1.4.2

Cancel the common factor.

$e = 0 - (\frac{4 \cdot 4}{9} \cdot \frac{1}{4})$

Step 6.4.2.1.4.3

Rewrite the expression.

$e = 0 - \frac{4}{9}$

Step 6.4.2.2

Subtract $\frac{4}{9}$ from $0$ .

$e = - \frac{4}{9}$

Step 6.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + \frac{2}{3})}^{2} - \frac{4}{9}$ .

${(y + \frac{2}{3})}^{2} - \frac{4}{9}$

Step 7

Substitute ${(y + \frac{2}{3})}^{2} - \frac{4}{9}$ for $y^{2} + \frac{4 y}{3}$ in the equation $x^{2} + y^{2} + \frac{x}{2} + \frac{4 y}{3} = - \frac{37}{144}$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} - \frac{4}{9} = - \frac{37}{144} + \frac{1}{16}$

Step 8

Move $- \frac{4}{9}$ to the right side of the equation by adding $\frac{4}{9}$ to both sides.

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{1}{16} + \frac{4}{9}$

Step 9

Simplify $- \frac{37}{144} + \frac{1}{16} + \frac{4}{9}$ .

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Step 9.1

Find the common denominator.

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Step 9.1.1

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{1}{16} \cdot \frac{9}{9} + \frac{4}{9}$

Step 9.1.2

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{9}{16 \cdot 9} + \frac{4}{9}$

Step 9.1.3

Multiply $\frac{4}{9}$ by $\frac{16}{16}$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{9}{16 \cdot 9} + \frac{4}{9} \cdot \frac{16}{16}$

Step 9.1.4

Multiply $\frac{4}{9}$ by $\frac{16}{16}$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{9}{16 \cdot 9} + \frac{4 \cdot 16}{9 \cdot 16}$

Step 9.1.5

Reorder the factors of $16 \cdot 9$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{9}{9 \cdot 16} + \frac{4 \cdot 16}{9 \cdot 16}$

Step 9.1.6

Multiply $9$ by $16$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{9}{144} + \frac{4 \cdot 16}{9 \cdot 16}$

Step 9.1.7

Multiply $9$ by $16$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = - \frac{37}{144} + \frac{9}{144} + \frac{4 \cdot 16}{144}$

Step 9.2

Combine the numerators over the common denominator.

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{- 37 + 9 + 4 \cdot 16}{144}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Multiply $4$ by $16$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{- 37 + 9 + 64}{144}$

Step 9.3.2

Add $- 37$ and $9$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{- 28 + 64}{144}$

Step 9.3.3

Add $- 28$ and $64$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{36}{144}$

Step 9.4

Cancel the common factor of $36$ and $144$ .

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Step 9.4.1

Factor $36$ out of $36$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{36 (1)}{144}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $36$ out of $144$ .

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{36 \cdot 1}{36 \cdot 4}$

Step 9.4.2.2

Cancel the common factor.

${(x + \frac{1}{4})}^{2} + {(y + \frac{2}{3})}^{2} = \frac{36 \cdot 1}{36 \cdot 4}$

Step 9.4.2.3

Rewrite the expression.

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