**Solving Word Problems with Nature of Roots of Quadratic Equation :**

In this section, we will learn, how to solve word problems with the concept nature of roots of quadratic equation.

**Example 1 :**

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 x be the breadth of the rectangular grove

length = 2 x

Area of rectangular grove = 800 m²

x (2x) = 800

2x² = 800

x² = 800/2

x² = 400

x = √400

x = 20

breadth of the rectangular grove = 20 m

length of rectangular grove = 2 (20) = 40 m

**Example 2 :**

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 "x" be the age of one friend

age of another friend = 20 - x

four years ago their ages was

x - 4 and 20 - x - 4

x - 4 and 16 - x

The product of their ages = 48

(x - 4) (16 - x) = 48

16 x - x² - 64 + 4 x = 48

- x²+ 20 x - 64 - 48 = 0

- x²+ 20 x - 112 = 0

x²- 20 x + 112 = 0

by comparing this equation with general form of quadratic equation we get,

a = 1 b = -20 and c = 112

Discriminant = b² - 4 a c

= (-20)² - 4 (1) (112)

= 400 - 448

= -48 < 0

Since discriminant is less than zero, there is no real roots for this equation. So, we can say the above situation is not possible.

**Example 3 :**

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

**Solution :**

perimeter of rectangular park = 80

2 (l + b) = 80

l + b = 40 -----(1)

Area of rectangular park = 400 m²

l x b = 400

b = 400/l ------(2)

Substitute (2) equation in the first equation

l + (400/l) = 40

(l² + 400)/l = 40

l² + 400 = 40 l

l² - 40 l + 400 = 0

l² - 20 l - 20 l + 400 = 0

l (l - 20) - 20 (l - 20) = 0

(l - 20) (l - 20) = 0

l = 20 m

b = 400/20

b = 20 m

So, the length of the rectangular park = 20 m and breadth of the rectangular park = 20 m.

After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems with nature of roots of a quadratic equation.

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