


Sam decides to build a square garden. If the area of the garden is 9x2 − 24x + 16 square feet, what is the length of one side of the garden?

To determine the length of one side of Sam's square garden, we need to express the area of the garden, given by \( 9x^2 - 24x + 16 \) square feet, as a perfect square trinomial.

The general form of a perfect square trinomial is:

\[
(ax + b)^2 = a^2x^2 + 2abx + b^2
\]

Let's see if we can factor \( 9x^2 - 24x + 16 \) into a square of a binomial.

1. **Identify the first and last terms:**
   - The first term is \( 9x^2 \), which can be written as \( (3x)^2 \).
   - The last term is \( 16 \), which can be written as \( 4^2 \).

2. **Check the middle term:**
   - The middle term in a perfect square trinomial is \( 2abx \).
   - Here, \( a = 3x \) and \( b = 4 \), so \( 2ab = 2(3x)(4) = 24x \).

The middle term is \( -24x \), which matches the given expression if we include a negative sign. This confirms that we can write the area as:

\[
9x^2 - 24x + 16 = (3x - 4)^2
\]

Therefore, the length of one side of the square garden is \( 3x - 4 \) feet.

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**Do you want more details or have any questions?**

### Related Questions:
1. How do you factor other quadratic expressions?
2. What is the significance of recognizing perfect square trinomials?
3. How do you determine if a quadratic expression can be factored into a perfect square?
4. What are other methods to solve for the side length of a square when given its area?
5. Can the length of a side be negative in any context?
6. How would you solve the equation if it wasn’t a perfect square trinomial?
7. How does this concept apply to solving real-world problems involving area?
8. What happens if you solve \( 3x - 4 = 0 \)? 

**Tip:** Always check your factorization by expanding the binomial to ensure it matches the original expression.

Learn how to calculate the length of one side of a square garden using algebraic methods. This problem involves factoring a perfect square trinomial to find the side length. Suitable for students in Grades 9-10.

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