Consider the quadratic equation  
x
2
+
12
x
+
47
 .

(A) Fill in the blank to complete the square:

x
2
+
12
x
+
47
=
⎛
⎝
x
2
+
12
x
+
(
2
)
2
⎞
⎠
−
(
2
)
2
+
47
=
(
x
2
+
12
x
+
)
+
(
−
+
47
)
=
(
x
+
)
2
+


Let's complete the square for the quadratic equation step by step.

We begin with the equation:

\[
x^2 + 12x + 47
\]

### Step 1: Find the number to complete the square.

To complete the square, we need to take half of the coefficient of \(x\) (which is 12), square it, and then add and subtract this number inside the equation.

Half of 12 is:

\[
\frac{12}{2} = 6
\]

Now, square that value:

\[
6^2 = 36
\]

So, we add and subtract 36 to the equation.

### Step 2: Rewrite the equation with the completed square.

\[
x^2 + 12x + 47 = (x^2 + 12x + 36) - 36 + 47
\]

### Step 3: Simplify.

\[
(x^2 + 12x + 36) - 36 + 47 = (x + 6)^2 + 11
\]

### Final result:

\[
x^2 + 12x + 47 = (x + 6)^2 + 11
\]

### Answer:

(A) Filling in the blanks:

\[
x^2 + 12x + 47 = (x^2 + 12x + (6)^2) - (6)^2 + 47 = (x^2 + 12x + 36) + (-36 + 47) = (x + 6)^2 + 11
\]

Do you want any further details or have questions?

Here are 5 related questions:
1. What is the vertex form of a quadratic equation?
2. How can completing the square help solve quadratic equations?
3. What is the geometric interpretation of completing the square?
4. Can completing the square be used for any quadratic equation?
5. How does completing the square relate to the quadratic formula?

**Tip:** Completing the square is a method that can be applied to any quadratic equation, and it also helps to derive the quadratic formula!

Consider the quadratic equation x^2 + 12x + 47.

(A) Fill in the blank to complete the square:

x^2 + 12x + 47 = (x^2 + 12x + (__)^2) - (__)^2 + 47 = (x^2 + 12x + __) + (−__ + 47) = (x + __)^2 + __

Learn how to complete the square for the quadratic equation x^2 + 12x + 47. This detailed solution guides students through key algebraic steps, including adding and subtracting the square of half the linear coefficient. Suitable for students in Grades 9-11.

Math Problem Statement

x 2 + 12 x + 47

⎛ ⎝ x 2 + 12 x + ( 2 ) 2 ⎞ ⎠ − ( 2 ) 2 + 47

( x 2 + 12 x + ) + ( − + 47 )

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