Find the Standard Form of the Parabola y=x^2+6x+9

$y = x^{2} + 6 x + 9$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 6 x + 9 = y$

Step 2

Subtract $9$ from both sides of the equation.

$x^{2} + 6 x = y - 9$

Step 3

Complete the square.

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

$a = 1$

$b = 6$

$c = 0 = y - 9$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 9$

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

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

Step 4

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

Add $9$ to both sides of the equation.

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

Combine the opposite terms in $y - 9 + 9$ .

Add $- 9$ and $9$ .

${(x + 3)}^{2} = y + 0$

Add $y$ and $0$ .

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

Find the Standard Form of the Parabola y=x^2+6x+9

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