Let the width of the rectangular field be $x$ meters. Since the field is 15 meters longer than it is wide, its length is $x + 15$ meters.

Step 1: Set up the area equation

The area of the rectangle is given as $800 \, \text{m}^2$. Using the formula for the area of a rectangle:

$\text{Area} = \text{Length} \times \text{Width}$

Substitute the expressions for length and width:

$x(x + 15) = 800$

Step 2: Formulate a quadratic equation

Expand the equation:

$x^2 + 15x = 800$

Rearrange into standard quadratic form:

$x^2 + 15x - 800 = 0$

Step 3: Solve the quadratic equation

To solve $x^2 + 15x - 800 = 0$, use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = 15$, and $c = -800$. Substituting these values:

$x = \frac{-15 \pm \sqrt{15^2 - 4(1)(-800)}}{2(1)}$

$x = \frac{-15 \pm \sqrt{225 + 3200}}{2}$

$x = \frac{-15 \pm \sqrt{3425}}{2}$

$x = \frac{-15 \pm 58.5}{2}$

Step 4: Calculate the two roots

$x = \frac{-15 + 58.5}{2} \quad \text{or} \quad x = \frac{-15 - 58.5}{2}$

$x = \frac{43.5}{2} = 21.75 \quad \text{or} \quad x = \frac{-73.5}{2} = -36.75$

Since $x$, the width, cannot be negative, we take $x = 21.75$.

Step 5: Find the length

The length of the field is:

$x + 15 = 21.75 + 15 = 36.75 \, \text{m}$

Final Answer:

The length of the field is 36.8 meters (to 1 decimal place).

---

Do you want further clarification or steps explained? Here are some related questions you might find helpful:

1. How do you derive the quadratic formula?
2. Can you provide alternative ways to solve quadratic equations?
3. How do you check if a quadratic equation has real roots?
4. Can you explain why negative roots are invalid in this problem?
5. How does rounding affect real-world applications?

Tip: Always verify your solution by substituting it back into the original equation to ensure accuracy.