Write in Standard Form x^2-14x+49+y^2+4y=0

$x^{2} - 14 x + 49 + y^{2} + 4 y = 0$

Step 1

Subtract $49$ from both sides of the equation.

$x^{2} - 14 x + y^{2} + 4 y = - 49$

Step 2

Complete the square for $x^{2} - 14 x$ .

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

$a = 1$

$b = - 14$

$c = 0$

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

$d = \frac{- 14}{2 \cdot 1}$

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

$d = \frac{2 \cdot - 7}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 7}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 7}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 7}{1}$

Divide $- 7$ by $1$ .

$d = - 7$

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

$e = 0 - \frac{{(- 14)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 14$ to the power of $2$ .

$e = 0 - \frac{196}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{196}{4}$

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Subtract $49$ from $0$ .

$e = - 49$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 7)}^{2} - 49$ .

${(x - 7)}^{2} - 49$

Step 3

Substitute ${(x - 7)}^{2} - 49$ for $x^{2} - 14 x$ in the equation $x^{2} - 14 x + y^{2} + 4 y = - 49$ .

${(x - 7)}^{2} - 49 + y^{2} + 4 y = - 49$

Step 4

Move $- 49$ to the right side of the equation by adding $49$ to both sides.

${(x - 7)}^{2} + y^{2} + 4 y = - 49 + 49$

Step 5

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

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

$a = 1$

$b = 4$

$c = 0$

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

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = 2$

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

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

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

Step 6

Substitute ${(y + 2)}^{2} - 4$ for $y^{2} + 4 y$ in the equation $x^{2} - 14 x + y^{2} + 4 y = - 49$ .

${(x - 7)}^{2} + {(y + 2)}^{2} - 4 = - 49 + 49$

Step 7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(x - 7)}^{2} + {(y + 2)}^{2} = - 49 + 49 + 4$

Step 8

Simplify $- 49 + 49 + 4$ .

Add $- 49$ and $49$ .

${(x - 7)}^{2} + {(y + 2)}^{2} = 0 + 4$

Add $0$ and $4$ .

${(x - 7)}^{2} + {(y + 2)}^{2} = 4$

Write in Standard Form x^2-14x+49+y^2+4y=0

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