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).