word problem involving completing the square

Here's a word problem that involves completing the square:

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**Problem:**

A farmer has a rectangular garden that is 6 meters longer than it is wide. The area of the garden is 160 square meters. Find the dimensions of the garden using the method of completing the square.

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**Solution:**

Let's denote the width of the garden by \( x \) meters. According to the problem, the length of the garden is \( x + 6 \) meters.

The area of the garden is given by the product of its length and width:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

Substituting the values we have:

\[
x(x + 6) = 160
\]

This expands to:

\[
x^2 + 6x = 160
\]

To solve this quadratic equation by completing the square, follow these steps:

1. **Move the constant to the other side:**

   \[
   x^2 + 6x = 160
   \]

2. **Add and subtract the square of half the coefficient of \( x \) (which is \( \frac{6}{2} = 3 \), so the square is \( 9 \)) to both sides:**

   \[
   x^2 + 6x + 9 = 160 + 9
   \]

   This simplifies to:

   \[
   (x + 3)^2 = 169
   \]

3. **Take the square root of both sides:**

   \[
   x + 3 = \pm \sqrt{169}
   \]

   \[
   x + 3 = \pm 13
   \]

4. **Solve for \( x \):**

   - \( x + 3 = 13 \) gives \( x = 10 \)
   - \( x + 3 = -13 \) gives \( x = -16 \) (which we discard since the width cannot be negative)

Thus, the width \( x \) of the garden is 10 meters.

Since the length is \( x + 6 \), the length is:

\[
10 + 6 = 16 \text{ meters}
\]

**Final Answer:**

The dimensions of the garden are 10 meters by 16 meters.

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Would you like more details on any part of this solution or have any questions?

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**Related Questions:**

1. How do you complete the square for a quadratic equation with a coefficient other than 1?
2. What is the geometric interpretation of completing the square?
3. Can completing the square be used to solve any quadratic equation?
4. How would the solution change if the garden's area was different?
5. What other methods can be used to solve quadratic equations?

**Tip:** When completing the square, always remember to add the same value to both sides of the equation to maintain equality.

Learn how to solve a word problem involving completing the square method in algebra. This problem deals with finding dimensions of a garden given its area and relationship between length and width. Suitable for high school students.

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