A square tile covers an area of (9x ^ 2 + 12x + 4) square units. Find the measure of each side of the tile.

The area of a square tile is given as \( 9x^2 + 12x + 4 \) square units. Since the area of a square is \( \text{side}^2 \), we need to express this quadratic expression as a perfect square.

1. The expression \( 9x^2 + 12x + 4 \) looks like a perfect square trinomial. Let's factor it.

\[
9x^2 + 12x + 4 = (3x + 2)^2
\]

2. Now that we have factored the quadratic, the side length of the square tile is the square root of the area:

\[
\text{side} = 3x + 2
\]

Thus, the measure of each side of the tile is \( 3x + 2 \) units.

Would you like more details on how to factor the quadratic expression or have other questions?

Here are 5 related questions to explore further:
1. How do you factor perfect square trinomials?
2. Can all quadratic expressions be factored easily?
3. What are some other methods to solve for the side of a square when given the area?
4. What is the geometric interpretation of completing the square?
5. How does this method generalize to other shapes?

**Tip:** When factoring quadratic expressions, always check for common patterns like perfect square trinomials or differences of squares to simplify your work.

A square tile covers an area of (9x^2 + 12x + 4) square units. Find the measure of each side of the tile.

Learn how to factor the quadratic expression 9x^2 + 12x + 4 to find the side length of a square tile. This problem covers algebraic concepts such as factoring perfect square trinomials and solving for side lengths. Suitable for students in Grades 8-10.

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