Write in Standard Form 144x^2+144y^2+72x+192y+37=0

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

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 $x^{2} + \frac{x}{2}$ .

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

$a = 1$

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

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Simplify the right side.

Simplify each term.

Simplify the numerator.

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

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

One to any power is one.

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

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

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

Multiply $4$ by $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $4$ by $4$ .

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

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

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

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 $y^{2} + \frac{4 y}{3}$ .

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

$a = 1$

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

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Simplify the right side.

Simplify each term.

Simplify the numerator.

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

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

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

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

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

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

Multiply $4$ by $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $4$ .

Factor $4$ out of $16$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Find the common denominator.

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

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

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

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

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

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

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

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

Reorder the factors of $16 \cdot 9$ .

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

Multiply $9$ by $16$ .

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

Multiply $9$ by $16$ .

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

Combine the numerators over the common denominator.

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

Simplify the expression.

Multiply $4$ by $16$ .

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

Add $- 37$ and $9$ .

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

Add $- 28$ and $64$ .

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

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

Factor $36$ out of $36$ .

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

Cancel the common factors.

Factor $36$ out of $144$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Write in Standard Form 144x^2+144y^2+72x+192y+37=0

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