9. Carly is solving quadratic equations by completing the square. She is solving the following problem, but is stuck at the next step. Fill in the next step for her. $$ \left.\begin{array}{l}x^{2}+2 x-22=8 \ x^{2}+2 x+\ldots \ldots \ldots=30+\ldots \ldots \ldots \ x^{2}+2 x+1=30+1 \ (x+\square\end{array} ight)^{2}=31 $$

Question

9. Carly is solving quadratic equations by completing the square. She is solving the following problem, but is stuck at the next step. Fill in the next step for her. $$ \left.\begin{array}{l}x^{2}+2 x-22=8 \ x^{2}+2 x+\ldots \ldots \ldots=30+\ldots \ldots \ldots \ x^{2}+2 x+1=30+1 \ (x+\square\end{array}ight)^{2}=31 $$

UpStudy AI · Accepted Answer

Carly is on the right track with completing the square for the equation:

$$
x^{2} + 2x - 22 = 8
$$

Let's go through the steps she has taken and complete the next step:

1. **Move the constant to the other side:**

$$
x^{2} + 2x = 30
$$

2. **Complete the square:**

To complete the square, take half of the coefficient of $$ x $$, which is $$ \frac{2}{2} = 1 $$, and then square it to get $$ 1 $$. Add this value to both sides of the equation:

$$
x^{2} + 2x + 1 = 30 + 1
$$

3. **Rewrite as a perfect square:**

$$
(x + 1)^2 = 31
$$

So, the next step that Carly needs to fill in is:

$$
(x + \boxed{1})^{2} = 31
$$

This indicates that the completed square is $$ (x + 1)^2 $$, and the next step involves taking the square root of both sides to solve for $$ x $$.

**Final Answer:**

$$
(x + \boxed{1})^{2} = 31
$$
$$ (x + 1)^2 = 31 $$

Carly is solving quadratic equations by completing the square. She is solving the following problem, but is stuck at the next step. Fill in the next step for her.

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