To solve the quadratic equation [tex]\(x^2 + 10x + \square = 11\)[/tex] by completing the square, we follow these steps:
1. Identify the coefficient of [tex]\(x\)[/tex]:
The coefficient [tex]\(b\)[/tex] of [tex]\(x\)[/tex] in the term [tex]\(x^2 + 10x\)[/tex] is [tex]\(10\)[/tex].
2. Calculate [tex]\((b/2)^2\)[/tex]:
To complete the square, we need to add and subtract [tex]\((b/2)^2\)[/tex]. Here, [tex]\(b = 10\)[/tex], so [tex]\((b/2)^2 = (10/2)^2 = 5^2 = 25\)[/tex].
3. Rewrite the equation:
We rewrite the equation by adding and subtracting [tex]\((b/2)^2\)[/tex], which is 25:
[tex]\[
x^2 + 10x + 25 = 11 + 25
\][/tex]
Simplifying the right side:
[tex]\[
x^2 + 10x + 25 = 36
\][/tex]
4. Express the left side as a square:
The left side can be written as a perfect square:
[tex]\[
(x + 5)^2 = 36
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Take the square root of both sides:
[tex]\[
x + 5 = \pm \sqrt{36}
\][/tex]
Since [tex]\(\sqrt{36} = 6\)[/tex], we have:
[tex]\[
x + 5 = 6 \quad \text{or} \quad x + 5 = -6
\][/tex]
6. Solve the resulting linear equations:
[tex]\[
\begin{align*}
x + 5 = 6 & \quad \Rightarrow \quad x = 6 - 5 = 1 \\
x + 5 = -6 & \quad \Rightarrow \quad x = -6 - 5 = -11
\end{align*}
\][/tex]
Therefore, the solutions to the equation are [tex]\(x = 1\)[/tex] and [tex]\(x = -11\)[/tex]. Thus, the answer is:
A. [tex]\( x = 1\)[/tex], and [tex]\(x = -11\)[/tex]
