CBSE Class 10 - Maths - CH4 Quadratic Equations (Ex 4.1)

Quadratic Equations

NCERT Exercise 4.1

Standard form of equation is 

ax² + bx + c = 0

where a, b and c are the co-efficients and a ≠ 0

Q1: Check whether the following are quadratic equations :
(i)   (x + 1)² = 2(x – 3)
(ii)  x² – 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3)
(iv) (x – 3)(2x +1) = x(x + 5)
(v)  (2x – 1)(x – 3) = (x + 5)(x – 1)
(vi) x² + 3x + 1 = (x – 2)²

Answer:


(i)   (x + 1)² = 2(x – 3)
⇒  x² + 2x + 1 = 3x - 6
⇒  x² + (2-3)x + 1 + 6 = 0
⇒  x² - x + 7 = 0
Since the equation is of the form ax² + bx + c = 0, it is a quadratic equation.

(ii) x² – 2x = (–2) (3 – x)
⇒  x²– 2x = (–2 × 3) + ( –2 × –x)
⇒x²– 2x = –6 + 2x
⇒  x²– 2x - 2x + 6 = 0
⇒  x²– 4x + 6 = 0
Since the equation is of the form ax² + bx + c = 0, it is a quadratic equation.

(iii) (x – 2)(x + 1) = (x – 1)(x + 3)
⇒ x² + x -2x - 2 = x² + 3x -x -3
⇒ x² - x - 2 = x² + 2x - 3
⇒ x² - x - 2 - x² - 2x + 3 = 0
⇒ -3x + 1 = 0
 ⇒ 3x - 1 = 0
Since the equation is NOT of the form ax² + bx + c = 0, it is NOT a quadratic equation.

(iv) (x – 3)(2x +1) = x(x + 5)
⇒ 2x² + x - 6x - 3 = x² + 5x
⇒ 2x² - 5x - 3 -  x² - 5x = 0
⇒ x²- 10x - 3 = 0
Since the equation is of the form ax² + bx + c = 0, it is a quadratic equation.

(v)  (2x – 1)(x – 3) = (x + 5)(x – 1)
⇒ 2x² -  6x  - 1x + 3 = x² - 1x + 5x - 5
⇒ 2x² - 7x + 3 = x²+ 4x - 5
⇒ 2x² - 7x + 3 - x² - 4x + 5 = 0
⇒ x² - 11x + 8 = 0
Since the equation is of the form ax² + bx + c = 0, it is a quadratic equation.

(vi) x² + 3x + 1 = (x – 2)²
⇒ x² + 3x + 1 = x² + 4 - 4x
⇒ x² + 3x + 1 - x² - 4 + 4x =0
⇒ 7x - 3 = 0
Since the equation is NOT of the form ax² + bx + c = 0, it is NOT a quadratic equation.

Q2: 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.

Answer: Let the breadth of the rectangular plot = b m
∴ Length of the plot = (2 × b + 1) m = 2b + 1
Area = length × breadth = 528 m²
⇒ (2b+1) × b = 528
⇒ 2b² + b =528⇒ 2b² + b  - 528 = 0

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

Answer: Let the first integer number be = p
The next consecutive positive integer =  p+1
Product = p × (p +1) = 306
⇒ p²+ p = 306
⇒ p²+ p - 206 = 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. 

Answer: Let Rohan's age = t years
Rohan's mother age = t + 26 years
3 years from now, Rohan's age = t + 3
Age of Rohan's mother after 3 years = t + 26 + 3 = t + 29
Product of their ages after three years = (t + 3)(t + 29) = 360
⇒ t² + 29t + 3t + 87 = 360
⇒ t² + 32t + 87 - 360 = 0
⇒ t² + 32t - 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.

Answer: Let the speed of the train = v km/h
∵ Distance = speed  × time

Time taken by  train (t) = 480/v  h
If speed decreases by 8 km/h, new speed = v - 8  km/h
Time taken to cover 480 km with new speed  = t + 3 hours = 480/v + 3 hours
∵ Distance = speed × time
⇒  (v - 8) × (480/v + 3) = 480
⇒  480 + 3v - 3840/v - 24 = 480

⇒  480 + 3v - 3840/v - 24 - 480 = 0
⇒ 3v - 3840/v - 24 = 0
⇒ 3v² - 3840 - 24v = 0
⇒ v² - 8v - 1280  = 0

Q3(FA Manual): Which of the following are quadratic equations?
(i) √x - x = 4
(ii) x + 1/x = 5
(iii) 7 v² = 49

Answer:
(i) √x - x = 4
⇒ x^1/2- x = 4
Since Since the equation is NOT of the form ax² + bx + c = 0, it is NOT a quadratic equation.

(ii) x + 1/x = 5
⇒ x + 1/x - 5 = 0
Multiply by x to both sides,
⇒ x² + 1 - 5x = 0
⇒ x² -5x + 1 = 0
Since the equation is of the form ax² + bx + c = 0, it is a quadratic equation.

(iii) 7 v² = 49
⇒ 7v² - 49 = 0
⇒  7v²+ 0v  - 49 = 0


Since the equation is of the form ax² + bx + c = 0, it is a quadratic equation.

[Note: Equation is quadratic, when the first term must be  ax² and a ≠ 0 but coefficients b and c can be zero].







.