Write in Standard Form 4x^2-24x+4y^2+72y=76

$4 x^{2} - 24 x + 4 y^{2} + 72 y = 76$

Divide both sides of the equation by $4$ .

$x^{2} - 6 x + y^{2} + 18 y = 19$

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

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

$a = 1$

$b = - 6$

$c = 0$

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

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

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

Factor $2$ out of $- 6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 3$ by $1$ .

$d = - 3$

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $1$ .

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

Divide $36$ by $4$ .

$e = 0 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Subtract $9$ from $0$ .

$e = - 9$

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

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

Substitute ${(x - 3)}^{2} - 9$ for $x^{2} - 6 x$ in the equation $x^{2} - 6 x + y^{2} + 18 y = 19$ .

${(x - 3)}^{2} - 9 + y^{2} + 18 y = 19$

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

${(x - 3)}^{2} + y^{2} + 18 y = 19 + 9$

Complete the square for $y^{2} + 18 y$ .

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

$a = 1$

$b = 18$

$c = 0$

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

$d = \frac{18}{2 \cdot 1}$

Cancel the common factor of $18$ and $2$ .

Factor $2$ out of $18$ .

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot 9}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{9}{1}$

Divide $9$ by $1$ .

$d = 9$

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

$e = 0 - \frac{18^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{324}{4}$

Divide $324$ by $4$ .

$e = 0 - 1 \cdot 81$

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Subtract $81$ from $0$ .

$e = - 81$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 9)}^{2} - 81$ .

${(y + 9)}^{2} - 81$

Substitute ${(y + 9)}^{2} - 81$ for $y^{2} + 18 y$ in the equation $x^{2} - 6 x + y^{2} + 18 y = 19$ .

${(x - 3)}^{2} + {(y + 9)}^{2} - 81 = 19 + 9$

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

${(x - 3)}^{2} + {(y + 9)}^{2} = 19 + 9 + 81$

Simplify $19 + 9 + 81$ .

Add $19$ and $9$ .

${(x - 3)}^{2} + {(y + 9)}^{2} = 28 + 81$

Add $28$ and $81$ .

${(x - 3)}^{2} + {(y + 9)}^{2} = 109$

Write in Standard Form 4x^2-24x+4y^2+72y=76

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