Find the Standard Form of the Parabola x^2-4x+4y+24=0

$x^{2} - 4 x + 4 y + 24 = 0$

Step 1

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

Subtract $4 y$ from both sides of the equation.

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

Subtract $24$ from both sides of the equation.

$x^{2} - 4 x = - 4 y - 24$

Step 2

Complete the square.

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

$a = 1$

$b = - 4$

$c = 0 = - 4 y - 24$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = - 4 y - 24$

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 $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

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

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

Multiply $4^{2}$ by $1$ .

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

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 ${(x - 2)}^{2} - 4$ .

${(x - 2)}^{2} - 4 = - 4 y - 24$

Step 3

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

Add $4$ to both sides of the equation.

${(x - 2)}^{2} = - 4 y - 24 + 4$

Add $- 24$ and $4$ .

${(x - 2)}^{2} = - 4 y - 20$

Step 4

Factor.

Factor $4$ out of $- 4 y - 20$ .

Factor $4$ out of $- 4 y$ .

${(x - 2)}^{2} = 4 (- y) - 20$

Factor $4$ out of $- 20$ .

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

Factor $4$ out of $4 (- y) + 4 (- 5)$ .

${(x - 2)}^{2} = 4 (- y - 5)$

Factor $- 1$ out of $- y$ .

${(x - 2)}^{2} = 4 (- (y) - 5)$

Rewrite $- 5$ as $- 1 (5)$ .

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

Factor $- 1$ out of $- (y) - 1 (5)$ .

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

Multiply $- 1$ by $4$ .

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

Find the Standard Form of the Parabola x^2-4x+4y+24=0

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