Find the Standard Form of the Parabola x^2+20x-4y+100=0

$x^{2} + 20 x - 4 y + 100 = 0$

Move all terms not containing $x$ to the right side of the equation.

Add $4 y$ to both sides of the equation.

$x^{2} + 20 x + 100 = 4 y$

Subtract $100$ from both sides of the equation.

$x^{2} + 20 x = 4 y - 100$

Complete the square.

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

$a = 1$

$b = 20$

$c = 0 = 4 y - 100$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = 4 y - 100$

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

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

$d = \frac{20}{2 \cdot 1}$

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $10$ by $1$ .

$d = 10$

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

$e = 0 - \frac{20^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $20$ to the power of $2$ .

$e = 0 - \frac{400}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{400}{4}$

Divide $400$ by $4$ .

$e = 0 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Subtract $100$ from $0$ .

$e = - 100$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 10)}^{2} - 100$ .

${(x + 10)}^{2} - 100 = 4 y - 100$

Move all terms not containing $x$ to the right side of the equation.

Add $100$ to both sides of the equation.

${(x + 10)}^{2} = 4 y - 100 + 100$

Combine the opposite terms in $4 y - 100 + 100$ .

Add $- 100$ and $100$ .

${(x + 10)}^{2} = 4 y + 0$

Add $4 y$ and $0$ .

${(x + 10)}^{2} = 4 y$

Find the Standard Form of the Parabola x^2+20x-4y+100=0

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