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The problem presented in your screenshot states:

- A rectangular garden is **30 ft longer than it is wide**.
- The area of the garden is **5,400 square feet**.

We need to find the **width** and **length** of the garden.

### Step-by-step solution:

Let the width of the garden be \( w \) feet.

Since the length is 30 feet longer than the width, the length will be \( w + 30 \) feet.

The area of a rectangle is given by:
\[
\text{Area} = \text{length} \times \text{width}
\]
Substituting the given values:
\[
5400 = w \times (w + 30)
\]

Expanding the equation:
\[
5400 = w^2 + 30w
\]

Rearranging the equation into standard quadratic form:
\[
w^2 + 30w - 5400 = 0
\]

### Solving the quadratic equation:
We can solve this quadratic equation using the quadratic formula:
\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( w^2 + 30w - 5400 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 30 \)
- \( c = -5400 \)

Substitute these into the quadratic formula and solve for \( w \).

Let me calculate this for you.The solutions to the quadratic equation are:

- \( w_1 = 60 \) (positive value)
- \( w_2 = -90 \) (negative value, which doesn't make sense in this context since width can't be negative).

Therefore, the width of the garden is \( 60 \) feet.

### Length:
The length is 30 feet longer than the width:
\[
\text{Length} = 60 + 30 = 90 \, \text{feet}
\]

### Final dimensions:
- **Width** = 60 feet
- **Length** = 90 feet

Would you like further clarification on any steps?

Here are some related questions you might be interested in:
1. How do you complete the square to solve quadratic equations?
2. What are other methods to solve quadratic equations?
3. How can you graphically solve a quadratic equation?
4. What are real-world applications of quadratic equations in geometry?
5. How do you derive the quadratic formula?

**Tip:** When solving a quadratic equation, always check both solutions, but discard any that don't make sense in the context (e.g., negative lengths in geometry).

A rectangular garden is 30 ft longer than it is wide. Its area is 5,400 ft². What are its dimensions (in ft)?

Learn how to solve for the dimensions of a rectangular garden that is 30 feet longer than it is wide, with an area of 5,400 square feet. The solution involves quadratic equations and key algebraic concepts suitable for students in Grades 9-10.

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