Word problem that includes the factoring special product with the solution

Here’s a word problem that involves factoring a special product:

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**Word Problem:**

A square garden has a length that is 10 meters more than the width. The area of the garden is 400 square meters. What are the dimensions of the garden?

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**Solution:**

1. Let the width of the garden be \( x \) meters.
2. The length of the garden is \( x + 10 \) meters.
3. The area of a rectangle (or square in this case) is given by \( \text{Area} = \text{Length} \times \text{Width} \).
4. Therefore, the equation for the area of the garden becomes:
   \[
   x(x + 10) = 400
   \]
5. Expand the equation:
   \[
   x^2 + 10x = 400
   \]
6. Rearrange the equation to set it equal to 0:
   \[
   x^2 + 10x - 400 = 0
   \]
7. Now, solve this quadratic equation by factoring. We use the **quadratic formula** since this does not factor easily:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 1 \), \( b = 10 \), and \( c = -400 \).
   
   \[
   x = \frac{-10 \pm \sqrt{(10)^2 - 4(1)(-400)}}{2(1)}
   \]
   \[
   x = \frac{-10 \pm \sqrt{100 + 1600}}{2}
   \]
   \[
   x = \frac{-10 \pm \sqrt{1700}}{2}
   \]
   \[
   x = \frac{-10 \pm 41.23}{2}
   \]
   \[
   x = \frac{-10 + 41.23}{2} \quad \text{or} \quad x = \frac{-10 - 41.23}{2}
   \]
   So, \( x = 15.615 \) or \( x = -25.615 \).

Since the width of the garden cannot be negative, we take \( x = 15.615 \).

Thus, the width is approximately **15.62 meters**, and the length is \( 15.62 + 10 = 25.62 \) meters.

**Final Answer:**
The dimensions of the garden are approximately **15.62 meters by 25.62 meters**.

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Would you like more details or have any questions?

Here are 5 related questions:

1. How would you solve this problem if the area was a different number, like 500 square meters?
2. What is the relationship between the length and width of the garden in this word problem?
3. Can you factor the quadratic equation \(x^2 + 10x - 400 = 0\) by completing the square instead of using the quadratic formula?
4. What if the problem involved a rectangular garden rather than a square one? How would that change the equation?
5. Can you create a similar word problem involving the factoring of a perfect square trinomial?

**Tip:** When solving quadratic equations, always check if factoring is possible first, but be ready to use the quadratic formula if needed.

A square garden has a length that is 10 meters more than the width. The area of the garden is 400 square meters. What are the dimensions of the garden?

Solve a quadratic equation involving the area of a square garden where the length is 10 meters more than the width. This problem demonstrates how to apply the quadratic formula and algebraic concepts, including factoring special products. Suitable for Grades 9-10.

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