Exercise 10.4.1

2. Represent the following situations in the form of quadratic equations.

(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Solution:

Let the breadth of the plot be x meters. Then, the length of the plot is 2x+1 meters.

The area of the plot is given by: Area=Length×Breadth ⇒ 528 = 2x2 + x.

Rearranging the equation to standard quadratic form: 2x2+x−528=0

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

Solution:

Let the first integer be x. Then, the next consecutive integer is x+1.

The product of these integers is given by:x(x+1)=306 x2 + x = 306.

Rearranging the equation to standard quadratic form: x2+x−306 = 0

(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.

Solution :

The product of their ages 3 years from now is given by:(x+3)(x+29)=360 ⇒ x2+32x+87=360

Rearranging the equation to standard quadratic form: x2+32x+87−360 = 0 ⇒ x2+32x−273 = 0.

(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:

Let the speed of the train be x km/h.

Time taken to travel 480 km at speed x is: 480xhrs.

If the speed had been x−8 km/h, the time taken would be:480x−8hrs.

Rearranging to form a quadratic equation:480x−8 - 480x = 3.

Combining the fractions on the left-hand side:3840xx−8 = 3.

Multiplying both sides by x(x-8): ⇒ 3840 = 3x(x−8) ⇒ 3840 = 3x2 - 24x.

Rearranging the equation to standard quadratic form: 3x2−24x−3840=0.