Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800\(m^2\). If so, find its length and breadth.

Ans.Let breadth of rectangular mango grove = x metres
Let length of rectangular mango grove = 2x metres

Area of rectangle = length × breadth = \(x × 2x = 2x^2 m^2\)

According to given condition:

\(\Rightarrow 2x^2 = 800\)
\(\Rightarrow 2x^2 - 800 = 0 \)
\(\Rightarrow x^2 - 400 = 0\)

Comparing equation \( x^2 - 400 = 0\) with general form of quadratic equation \(ax^2 + bx + c = 0\) ,
we get a = 1, b = 0 and c = -400

Discriminant = \( b^2 -4ac \)
\( = (0)^2 - 4 (1) (-400) \)
\(= 1600\)

Discriminant is greater than 0 means that equation has two distinct real roots.

Therefore, it is possible to design a rectangular grove.

Applying quadratic formula,\(x = {{-b ± \sqrt{b^2 - 4ac}} \over {2a}}\) to solve equation,

\(x = {{0 ± \sqrt{1600}} \over {2(1)}}\) = ± {{40} \over {2} } = ± 20\)
x = 20, -20

We discard negative value of x because breadth of rectangle cannot be in negative.

Therefore, x = breadth of rectangle = 20 metres
Length of rectangle = 2x = 40 metres