Exercise 4.3

Find the nature of the roots of the following quadratic equations. If the real root exist, find them:

Find the values of k for each of the following quadratic equations, so that they have two equal roots.

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2 ? If so, find its length and breadth.

Solution

Let the breadth of the given rectangle be x m.

Therefore, the length will be 2x m.

Area of a rectangle is given by length × breadth

800 = (x) × (2x) [ Since area is given as 800 m2]

2x2 =

x2 = 8002

x2 =

x2 - 400 =

Discriminant of a quadratic equation ax2 + bx + c = 0 is b2 - .

Comparing x2 - 400 = 0 with ax2 + bx + c = 0 we have,

a = , b = , c =

b2 - 4ac = 02 - 4 × 1 ×

= > 0

As the discriminant is greater than 0, it is possible to have real distinctno distinct roots.

Hence, yesno, it is possible to design a mango grove.

x2 - 400 = 0

x2 =

x = ±

The value of x can’t be a negativepositive value as it represents the breadth of the rectangle.

Therefore, x = 20 m

Length = 2x = 2(20) = m

Breadth = x = m

Thus, it is possible to design a mango grove with length 40 m and breadth 20 m.

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Solution

Let the age of the first friend be x years.

According to the question, age of the second friend will be 20 - age of the first friend = 20 - x

Four years ago, age of the first friend = x -

Four years ago, age of the second friend = - x -

Product of their ages: (x - 4)(20 - x - 4) =

(x - 4)( - x) = 48

x - x2 - + 4x = 48

- x2 + x - 64 = 48

- x2 + 20x - 64 - = 0

- x2 + 20x - = 0

x2 - 20x + = 0

Let’s find the discriminant: b2 - 4ac

a = , b = , c =

b2 - 4ac = −202 - 4(1)(112)

= -

=

b2- 4ac <> 0

Therefore, there are no real roots. So, this situation is not possible.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.

Solution

Consider a rectangular park with length as 'l' and breadth as 'b' respectively.

Perimeter of a rectangle = 2(l + b) = ....(1)

Area of a rectangle = l × b = ....(2)

2(l + b) = 80

(l + b) =

l = 40 -

Substituting the value of l = 40 - b in equation (2)

(40 - b)(b) =

b - b2 = 400

40b - b2 - = 0

b2 - b + 400 = 0

Let’s find the discriminant: b2 - 4ac

a = , b = , c =

b2 - 4ac = −402 - 4(1)(400)

= -

=

Since, the value of the discriminant is 0, thus we can have two equaldistinct and realno real roots.

Therefore, it is possible to design a rectangular park with the given condition.

x = −b±b2−4ac2a

= (- b ± 0)2a

= -(- 40)2(1)

= 402

=

So, breadth of the rectangle is b = 20 m and its length is l = 40 - b = 20 m