To solve this problem, let's follow the steps methodically:
1. Define the variables:
- Let the breadth of the garden be [tex]\( b \)[/tex] meters.
- Given that the length of the garden is 5 meters more than its breadth, we can express the length as [tex]\( l = b + 5 \)[/tex] meters.
2. Write the equation for the area:
- The area of a rectangle is given by the product of its length and breadth. Therefore, the area [tex]\( A \)[/tex] can be written as:
[tex]\[
A = l \times b
\][/tex]
- We know the area is 500 square meters. Substituting [tex]\( l \)[/tex] and [tex]\( A \)[/tex] into the equation, we get:
[tex]\[
500 = (b + 5) \times b
\][/tex]
3. Solve the quadratic equation:
- Expanding the right-hand side, we get:
[tex]\[
500 = b^2 + 5b
\][/tex]
- Rearranging this equation to a standard quadratic form, we obtain:
[tex]\[
b^2 + 5b - 500 = 0
\][/tex]
4. Find the roots of the quadratic equation:
- Solving the quadratic equation [tex]\( b^2 + 5b - 500 = 0 \)[/tex] gives us the solutions for [tex]\( b \)[/tex]. The solutions to this equation are:
[tex]\[
b = -25 \quad \text{or} \quad b = 20
\][/tex]
5. Choose the positive solution:
- Since the breadth of a garden cannot be negative, we discard [tex]\( b = -25 \)[/tex] and select [tex]\( b = 20 \)[/tex] meters.
6. Calculate the length:
- Using the breadth [tex]\( b = 20 \)[/tex] meters, we find the length [tex]\( l \)[/tex] as:
[tex]\[
l = b + 5 = 20 + 5 = 25 \text{ meters}
\][/tex]
7. Conclusion:
- Therefore, the breadth of the rectangular garden is [tex]\( 20 \)[/tex] meters, and the length is [tex]\( 25 \)[/tex] meters.
