# Represent the following situations in the form of quadratic equations:

(i) The area of a rectangular plot is 528 m^{2}. The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot

(ii) The product of two consecutive positive integers is 306. We need to find the integers

(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train

**Solution:**

(i) We know that the area of a rectangle can be expressed as the product of its length and breadth.

Since we don’t know the length and breadth of the given rectangle, we assume the breadth of the plot to be a variable (x meters). Then, we use the given relationship between length and breadth: length = 1 + 2 times breadth.

Therefore, Length = 2x + 1

Area of rectangle = Length × Breadth

Breadth = x

Length = 2x + 1

Area of Rectangular Plot = Length × Breadth = 528 m^{2}

(2x + 1) × (x) = 528

2x^{2} + x = 528

2x^{2} + x - 528 = 0

Thus, quadratic equation is 2x^{2} + x - 528 = 0 , where x is the breadth of the rectangular plot.

(ii) Let the first integer be x.

Since the integers are consecutive, the next integer is (x + 1).

It is given that

First integer × Next integer = 306

Therefore,

x (x + 1) = 306

x (x +1) = 306

x^{2} + x = 306

x^{2} + x - 306 = 0

Thus, quadratic equation is x^{2} + x - 306 = 0. (Where x is the first integer)

(iii) Let assume that Rohan’s present age is x years. Then, from the first condition, the mother’s age is (x + 26) years.

Three years from now, Rohan’s age will be (x + 3) and Rohan’s mother age will be (x + 3) + 26 = (x + 29). The product of their ages is 360.

Therefore,

(x + 3) × (x + 29) = 360

x^{2} + 29x + 3x + 87 = 360

x^{2} + 32x + 87 - 360 = 0

x^{2} + 32x - 273 = 0

Thus, quadratic equation is x^{2} + 32x - 273 = 0 . Where x is the present age of Rohan.

(iv) Distance is equal to speed multiplied by time. Let the speed be s km/h and time be t hours.

D = st

480 = st

t = 480/s …(i)

As per the given conditions, for the same distance covered at a speed reduced by 8 km/h, the time taken would have increased by 3 hours.

Therefore,

(s - 8)(t + 3) = 480

st + 3s - 8t - 24 = 480

480 + 3s - 8 (480/s) - 24 = 480 [From …(i)]

3s -3840/s - 24 = 0

3s (s) - 3840 - 24 (s) = 0

3s^{2} - 24s - 3840 = 0

(3s^{2} - 24s - 3840)/3 = 0

s^{2} - 8s - 1280 = 0

Thus, the quadratic equation is s^{2} - 8s - 1280 = 0, where s is the speed of the train.

**Video Solution:**

## Represent the following situation in the form of quadratic equations: (i) The area of a rectangular plot is 528 m². The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot. (ii) The product of two consecutive positive integers is 306. We need to find the integers. (iii) Rohan’s mother is 26 years older than The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age. (iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

### NCERT Solutions Class 10 Maths Chapter 4 Exercise 4.1 Question 2 - Chapter 4 Exercise 4.1 Question 2:

The quadratic equations for i), ii), iii) and iv) are i) 2x^{2} + x − 528 = 0 , where x is the breadth of the rectangular plot , ii) x^{2} + x − 306 = 0 where x is a positive integer, iii) x^{2} + 32x − 273 = 0 where x is the present age of Rohan and, iv) S^{2} − 8S − 1280 = 0 where s is the speed of the train.