https://file.aiquickdraw.com/mathbot/file/1725846039445kqjg38f5.jpg

The question in your image is asking you to complete the expressions so they form "trinomios cuadrados perfectos" (perfect square trinomials). To solve this, we follow a specific rule for perfect square trinomials:

For an expression like \( x^2 + bx + c \) to be a perfect square trinomial, the value of \( c \) must be \( \left( \frac{b}{2} \right)^2 \).

Let's solve each expression from the image:

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### a. \( x^2 + \_ + 49 \)
For this to be a perfect square, the middle term should be \( 2 \times x \times 7 = 14x \), so the missing term is **14x**.

### b. \( x^2 - \_ + 1 \)
For this to be a perfect square, the middle term should be \( 2 \times x \times 1 = 2x \), so the missing term is **-2x**.

### c. \( x^2 + 10x + \_ \)
Here \( b = 10 \). To complete the square:
\[
\left( \frac{10}{2} \right)^2 = 25
\]
So the missing term is **25**.

### d. \( x^2 + \_ + 36 \)
For this to be a perfect square, the middle term should be \( 2 \times x \times 6 = 12x \), so the missing term is **12x**.

### e. \( x^2 - 6x + \_ \)
Here \( b = -6 \). To complete the square:
\[
\left( \frac{-6}{2} \right)^2 = 9
\]
So the missing term is **9**.

### f. \( x^2 - 16x + \_ \)
Here \( b = -16 \). To complete the square:
\[
\left( \frac{-16}{2} \right)^2 = 64
\]
So the missing term is **64**.

### g. \( x^2 + 18x + \_ \)
Here \( b = 18 \). To complete the square:
\[
\left( \frac{18}{2} \right)^2 = 81
\]
So the missing term is **81**.

### h. \( x^2 + \_ + 121 \)
For this to be a perfect square, the middle term should be \( 2 \times x \times 11 = 22x \), so the missing term is **22x**.

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Would you like any further explanation on how perfect square trinomials work or how to complete the square?

Here are 5 related questions to practice:
1. How do you factor a perfect square trinomial?
2. What is the general formula for completing the square?
3. How would you solve \( x^2 + 12x + 36 = 0 \)?
4. Can a trinomial with a negative constant be a perfect square?
5. How can completing the square help solve quadratic equations?

**Tip:** Completing the square is a method often used to derive the quadratic formula!

Complete the expression with the necessary term so that it becomes a perfect square trinomial.

Learn how to complete the square for different quadratic expressions to form perfect square trinomials. This problem focuses on the rule (b/2)^2 to identify the missing terms for expressions like x^2 + bx + c. Suitable for students in Grades 8-10.

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