Write in Standard Form x^2+y^2-x+10y+25=0

$x^{2} + y^{2} - x + 10 y + 25 = 0$

Step 1

Subtract $25$ from both sides of the equation.

$x^{2} + y^{2} - x + 10 y = - 25$

Step 2

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

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

$a = 1$

$b = - 1$

$c = 0$

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

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

Simplify the right side.

Cancel the common factor of $- 1$ and $1$ .

Rewrite $- 1$ as $- 1 (1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

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

Multiply $4$ by $1$ .

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

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

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

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 3

Substitute ${(x - \frac{1}{2})}^{2} - \frac{1}{4}$ for $x^{2} - x$ in the equation $x^{2} + y^{2} - x + 10 y = - 25$ .

${(x - \frac{1}{2})}^{2} - \frac{1}{4} + y^{2} + 10 y = - 25$

Step 4

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

${(x - \frac{1}{2})}^{2} + y^{2} + 10 y = - 25 + \frac{1}{4}$

Step 5

Complete the square for $y^{2} + 10 y$ .

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

$a = 1$

$b = 10$

$c = 0$

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

$d = \frac{10}{2 \cdot 1}$

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

$d = \frac{2 \cdot 5}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 5}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{5}{1}$

Divide $5$ by $1$ .

$d = 5$

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

$e = 0 - \frac{10^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Subtract $25$ from $0$ .

$e = - 25$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 5)}^{2} - 25$ .

${(y + 5)}^{2} - 25$

Step 6

Substitute ${(y + 5)}^{2} - 25$ for $y^{2} + 10 y$ in the equation $x^{2} + y^{2} - x + 10 y = - 25$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} - 25 = - 25 + \frac{1}{4}$

Step 7

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = - 25 + \frac{1}{4} + 25$

Step 8

Simplify $- 25 + \frac{1}{4} + 25$ .

Find the common denominator.

Write $- 25$ as a fraction with denominator $1$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25}{1} + \frac{1}{4} + 25$

Multiply $\frac{- 25}{1}$ by $\frac{4}{4}$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25}{1} \cdot \frac{4}{4} + \frac{1}{4} + 25$

Multiply $\frac{- 25}{1}$ by $\frac{4}{4}$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25 \cdot 4}{4} + \frac{1}{4} + 25$

Write $25$ as a fraction with denominator $1$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25 \cdot 4}{4} + \frac{1}{4} + \frac{25}{1}$

Multiply $\frac{25}{1}$ by $\frac{4}{4}$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25 \cdot 4}{4} + \frac{1}{4} + \frac{25}{1} \cdot \frac{4}{4}$

Multiply $\frac{25}{1}$ by $\frac{4}{4}$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25 \cdot 4}{4} + \frac{1}{4} + \frac{25 \cdot 4}{4}$

Combine the numerators over the common denominator.

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 25 \cdot 4 + 1 + 25 \cdot 4}{4}$

Simplify each term.

Multiply $- 25$ by $4$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 100 + 1 + 25 \cdot 4}{4}$

Multiply $25$ by $4$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 100 + 1 + 100}{4}$

Simplify by adding numbers.

Add $- 100$ and $1$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{- 99 + 100}{4}$

Add $- 99$ and $100$ .

${(x - \frac{1}{2})}^{2} + {(y + 5)}^{2} = \frac{1}{4}$

Write in Standard Form x^2+y^2-x+10y+25=0

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