Let the width of the rectangular garden be $w$ meters. Since the length is 3 meters longer than the width, the length can be expressed as $w + 3$ meters.

The area of a rectangle is given by the formula: $\text{Area} = \text{Length} \times \text{Width}$ Substituting the values into the formula: $40 = (w + 3) \times w$

Expanding the equation: $40 = w^2 + 3w$

Rearranging the equation to form a quadratic equation: $w^2 + 3w - 40 = 0$

Now, we solve this quadratic equation using the quadratic formula: $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $w^2 + 3w - 40 = 0$, the coefficients are:

• $a = 1$
• $b = 3$
• $c = -40$

Substituting these into the quadratic formula: $w = \frac{-3 \pm \sqrt{3^2 - 4(1)(-40)}}{2(1)}$ $w = \frac{-3 \pm \sqrt{9 + 160}}{2}$ $w = \frac{-3 \pm \sqrt{169}}{2}$ $w = \frac{-3 \pm 13}{2}$

This gives two possible solutions: $w = \frac{-3 + 13}{2} = \frac{10}{2} = 5 \quad \text{(valid solution)}$ $w = \frac{-3 - 13}{2} = \frac{-16}{2} = -8 \quad \text{(not valid, since width cannot be negative)}$

Thus, the width of the garden is $5$ meters. The length is: $w + 3 = 5 + 3 = 8 \text{ meters}$

Therefore, the dimensions of the garden are:

• Width: 5 meters
• Length: 8 meters

Would you like further details on the steps?

Here are some related questions to deepen your understanding:

1. How do you derive the quadratic formula?
2. How can you verify the solution for the area using the found dimensions?
3. What if the length was 5 meters longer than the width? How would that change the solution?
4. How do you solve quadratic equations by factoring instead of the quadratic formula?
5. How can you apply these principles to other geometric shapes?

Tip: When solving quadratic equations, always check if the discriminant ($b^2 - 4ac$) is positive, zero, or negative to determine the nature of the roots.