Represent the following situations in the form of quadratic equations

Question

(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.

(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.

Answer 1

(i) Let the breadth = x metres

    Length = 2 (Breadth) + 1

    Length = (2x + 1) metres

    Since Length × Breadth = Area

    ∴ (2x + 1) × x = 528

    2x² + x = 528

    2x² + x – 528 = 0

    Thus, the required quadratic equation is

    2x² + x – 528 = 0

(ii) Let the two consecutive numbers be x and (x + 1).

     ∵ Product of the numbers = 306

     ∴ x (x + 1) = 306

    ⇒ x² + x = 306

    ⇒ x² + x – 306 = 0

    Thus, the required equdratic equation is

   x² + x – 306 = 0

(iii) Let the present age = x

     ∴ Mother’s age = (x + 26) years

     After 3 years

    His age = (x + 3) years

    Mother’s age = [(x + 26) + 3] years

    = (x + 29) years

   According to the condition,

     

   ⇒ (x + 3) × (x + 29) = 360

   ⇒ x² + 29x + 3x + 87 = 360

  ⇒ x² + 29x + 3x + 87 – 360 = 0

 ⇒ x² + 32x – 273 = 0

 Thus, the required quadratic equation is

  x² + 32x – 273 = 0

(iv) Let the speed of the tram = u km/hr

    Distance covered = 480 km

     Time taken = Distance + Speed

    = (480 ÷ u) hours

    = 480/u hours

   In second case,

    Speed = (u – 8) km/ hour

     

   According to the condition,

  

 ⇒ 480u – 480(u – 8) = 3u(u – 8)

⇒ 480u – 480u + 3840 = 3u² – 24u

⇒ 3840 – 3u² + 24u = 0

⇒ 1280 – u² + 8u = 0

⇒ –1280 + u² – 8u = 0

⇒ u² – 8u – 1280 = 0

Thus, the required quadratic equation is

 u² – 8u – 1280 = 0