Nikki's den is 8 feet longer than it is wide. If the​ den's area is 128 square​ feet, what are the dimensions of the​ room?

Let's define the dimensions of the den. Let:

- \( w \) represent the width of the room.
- Then the length of the room will be \( w + 8 \) feet (since the length is 8 feet longer than the width).

The area of a rectangle is given by:
\[
\text{Area} = \text{length} \times \text{width}
\]

Substituting the given values:
\[
128 = w \times (w + 8)
\]

We now have a quadratic equation:
\[
w^2 + 8w - 128 = 0
\]

To solve this quadratic equation, we can either factor it, complete the square, or use the quadratic formula. We'll use the quadratic formula:
\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 8 \), and \( c = -128 \).

Let's calculate the solution.The two possible solutions for the width \( w \) are 8 feet and -16 feet. Since the width cannot be negative, the width of the room must be 8 feet.

Now, since the length is 8 feet longer than the width, the length is:
\[
\text{Length} = 8 + 8 = 16 \text{ feet}
\]

Thus, the dimensions of the den are:
- Width: 8 feet
- Length: 16 feet

Would you like any more details or further explanation?

Here are some related questions:
1. How do you solve a quadratic equation using factoring?
2. What is the importance of the discriminant in the quadratic formula?
3. How can you check your solution for a quadratic word problem?
4. What are other methods to solve a quadratic equation besides the quadratic formula?
5. How do changes in dimensions affect the area of a rectangle?

**Tip:** When solving quadratic equations, always check both possible solutions to ensure they make sense in the context of the problem.

Learn how to solve a word problem involving quadratic equations and area. Nikki's den is 8 feet longer than its width, and the area is 128 square feet. Follow a step-by-step process using algebra and the quadratic formula to find the dimensions. Ideal for students in Grades 8-10.

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