# Ex.4.4 Q3 Quadratic Equations Solutions - NCERT Maths Class 10

## Question

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

## Text Solution

**What is known?**

i) Mango groove length is twice its breadth.

ii) Area \(=800\,\rm{m}^².\)

**What is Unknown?**

Finding the possibility of mango groove and if possible length and breadth.

**Reasoning:**

Let the breadth of rectangle \(x\,\rm{ m.}\)

\(\text {Length}= 2x\,\rm{m}\)

\(\text{Area}= \rm{Length} × \rm{Breadth}\)

** Steps:**

\(\text{Area}=\rm{ Length} × \rm{Breadth} \)

\[\begin{align}800 &= x \times 2x\\2{x^2} &= 800\\{x^2} &= \frac{{800}}{2}\\{x^2} &= 400\\ {x^2} - 400 &= 0\end{align}\]

Discriminant of a quadratic equation is \(b² - 4ac.\) Comparing \({x^2} - 400 = 0\) with \(ax^{2}+bx+c=0\):

\(a = 1,\, b = 0, \,c = - 400\)

\[\begin{align}{b^2} - 4ac &= {{(0)}^2} - 4(1)( - 400)|\\ &= + 1600 > 0 \end{align}\]

\(\therefore\) Yes, it is possible to design a mango groove.

\[\begin{align}{x^2} - 400 &= 0\\{x^2} &= 400\\x\, &= \pm 20\end{align}\]

Value of *\(x\)* can’t be negative value as it represents the breadth of the rectangle.

\(\therefore\; x = 20\,\rm{m}\)

So, yes, possible to design the mango groove

Length \(= 2x = 2(20) = 40\,\rm{m} \)

Breadth \(= x = 20\,\rm{m}\)