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.