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fill in the blanks to complete the factorised form

The given equation is:

\[
x^2 + 6x + 9 = (x + \square)^2
\]

To fill in the blank, we need to recognize that the left-hand side of the equation is a perfect square trinomial. A perfect square trinomial has the form:

\[
a^2 + 2ab + b^2 = (a + b)^2
\]

Here, \(x^2 + 6x + 9\) can be rewritten as:

\[
(x)^2 + 2(x)(3) + (3)^2 = (x + 3)^2
\]

Thus, the correct number to fill in the blank is 3. The completed equation is:

\[
x^2 + 6x + 9 = (x + 3)^2
\]

Would you like more details, or do you have any questions? Here are some related questions to consider:

1. How do you recognize a perfect square trinomial?
2. Can you factorize other quadratic expressions like \(x^2 + 8x + 16\)?
3. What is the general form of a perfect square trinomial?
4. How do you complete the square for expressions that are not perfect squares?
5. What are the applications of factoring quadratics in solving equations?

**Tip:** Always check if a quadratic expression can be factored as a perfect square trinomial to simplify solving equations.

Learn how to complete the square for the quadratic equation x^2 + 6x + 9 = (x + 3)^2 with a detailed step-by-step solution. This problem demonstrates the concept of a perfect square trinomial and its application in algebraic equations. Suitable for students in Grades 9-12.

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