April 10, 2026
No Comments
Chapter 4: Quadratic Equations
Complete NCERT Solutions | CBSE Class 10 Mathematics (2025-26)
Class 10- Science Solutions
Class 10- English Solutions
Class 10- Maths Solutions
Class 10- Social Science Solutions
Class 10 (CBSE) - SYLLABUS
Class 10 (CBSE) - PYQ
Exercise 4.1 (Page 73)
12 marks each
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
(vii) (x + 2)³ = 2x(x² – 1)
(viii) x³ – 4x² – x + 1 = (x – 2)³
(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
(vii) (x + 2)³ = 2x(x² – 1)
(viii) x³ – 4x² – x + 1 = (x – 2)³
Answer:
(i) x² + 2x + 1 = 2x – 6 → x² + 7 = 0 → Yes, quadratic
(ii) x² – 2x = –6 + 2x → x² – 4x + 6 = 0 → Yes, quadratic
(iii) x² – x – 2 = x² + 2x – 3 → –3x + 1 = 0 → Not quadratic (linear)
(iv) 2x² – 5x – 3 = x² + 5x → x² – 10x – 3 = 0 → Yes, quadratic
(v) 2x² – 7x + 3 = x² + 4x – 5 → x² – 11x + 8 = 0 → Yes, quadratic
(vi) x² + 3x + 1 = x² – 4x + 4 → 7x – 3 = 0 → Not quadratic
(vii) x³ + 6x² + 12x + 8 = 2x³ – 2x → 0 = x³ – 6x² – 14x – 8 → Not quadratic
(viii) x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8 → 2x² – 13x + 9 = 0 → Yes, quadratic
(i) x² + 2x + 1 = 2x – 6 → x² + 7 = 0 → Yes, quadratic
(ii) x² – 2x = –6 + 2x → x² – 4x + 6 = 0 → Yes, quadratic
(iii) x² – x – 2 = x² + 2x – 3 → –3x + 1 = 0 → Not quadratic (linear)
(iv) 2x² – 5x – 3 = x² + 5x → x² – 10x – 3 = 0 → Yes, quadratic
(v) 2x² – 7x + 3 = x² + 4x – 5 → x² – 11x + 8 = 0 → Yes, quadratic
(vi) x² + 3x + 1 = x² – 4x + 4 → 7x – 3 = 0 → Not quadratic
(vii) x³ + 6x² + 12x + 8 = 2x³ – 2x → 0 = x³ – 6x² – 14x – 8 → Not quadratic
(viii) x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8 → 2x² – 13x + 9 = 0 → Yes, quadratic
23 marks each
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. Find the length and breadth.
(ii) The product of two consecutive positive integers is 306. 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. Find their present ages.
(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. Find the original speed of the train.
(i) The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. Find the length and breadth.
(ii) The product of two consecutive positive integers is 306. 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. Find their present ages.
(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. Find the original speed of the train.
Answer:
(i) Let breadth = x m, length = (2x + 1) m. Area = x(2x+1) = 528 → 2x² + x – 528 = 0
(ii) Let integers be x and x+1. x(x+1) = 306 → x² + x – 306 = 0
(iii) Let Rohan's age = x, mother's age = x+26. After 3 years: (x+3)(x+29) = 360 → x² + 32x – 273 = 0
(iv) Let original speed = x km/h. Time = 480/x. New speed = x–8, time = 480/(x–8). Equation: 480/(x–8) – 480/x = 3 → x² – 8x – 1280 = 0
(i) Let breadth = x m, length = (2x + 1) m. Area = x(2x+1) = 528 → 2x² + x – 528 = 0
(ii) Let integers be x and x+1. x(x+1) = 306 → x² + x – 306 = 0
(iii) Let Rohan's age = x, mother's age = x+26. After 3 years: (x+3)(x+29) = 360 → x² + 32x – 273 = 0
(iv) Let original speed = x km/h. Time = 480/x. New speed = x–8, time = 480/(x–8). Equation: 480/(x–8) – 480/x = 3 → x² – 8x – 1280 = 0
Exercise 4.2 (Page 76) – Factorization
13 marks each
Find the roots of the following quadratic equations by factorisation:
(i) x² – 3x – 10 = 0 (ii) 2x² + x – 6 = 0 (iii) √2 x² + 7x + 5√2 = 0
(iv) 2x² – x + 1/8 = 0 (v) 100x² – 20x + 1 = 0
(i) x² – 3x – 10 = 0 (ii) 2x² + x – 6 = 0 (iii) √2 x² + 7x + 5√2 = 0
(iv) 2x² – x + 1/8 = 0 (v) 100x² – 20x + 1 = 0
Answer:
(i) (x – 5)(x + 2) = 0 → Roots: 5, –2
(ii) (2x – 3)(x + 2) = 0 → Roots: 3/2, –2
(iii) (√2 x + 5)(x + √2) = 0 → Roots: –5/√2, –√2
(iv) (2x – 1/2)(x – 1/4) = 0 → Roots: 1/4, 1/4
(v) (10x – 1)² = 0 → Roots: 1/10, 1/10
(i) (x – 5)(x + 2) = 0 → Roots: 5, –2
(ii) (2x – 3)(x + 2) = 0 → Roots: 3/2, –2
(iii) (√2 x + 5)(x + √2) = 0 → Roots: –5/√2, –√2
(iv) (2x – 1/2)(x – 1/4) = 0 → Roots: 1/4, 1/4
(v) (10x – 1)² = 0 → Roots: 1/10, 1/10
23 marks
Solve: (i) x² – 45x + 324 = 0 (ii) x² – 55x + 750 = 0
Answer: (i) 12, 33 (ii) 25, 30
33 marks
Find two numbers whose sum is 27 and product is 182.
Answer: 13 and 14
43 marks
Find two consecutive positive integers, sum of whose squares is 365.
Answer: 13 and 14
53 marks
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Answer: Base = 12 cm, Altitude = 5 cm
63 marks
A cottage industry produces a certain number of pottery articles in a day. It was observed that on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹90, find the number of articles produced and the cost of each article.
Answer: Number of articles = 6, Cost per article = ₹15
Exercise 4.3 (Page 87) – Quadratic Formula
13 marks each
Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x² – 7x + 3 = 0 (ii) 2x² + x – 4 = 0 (iii) 4x² + 4√3 x + 3 = 0 (iv) 2x² + x + 4 = 0
(i) 2x² – 7x + 3 = 0 (ii) 2x² + x – 4 = 0 (iii) 4x² + 4√3 x + 3 = 0 (iv) 2x² + x + 4 = 0
Answer:
(i) Roots: 3, 1/2
(ii) Roots: (–1 ± √33)/4
(iii) Roots: –√3/2, –√3/2
(iv) No real roots (discriminant negative)
(i) Roots: 3, 1/2
(ii) Roots: (–1 ± √33)/4
(iii) Roots: –√3/2, –√3/2
(iv) No real roots (discriminant negative)
23 marks each
Find the roots of the quadratic equations given in Q1 above by applying the quadratic formula.
Answer:
(i) x = [7 ± √(49 – 24)]/4 = [7 ± 5]/4 → 3, 1/2
(ii) x = [–1 ± √(1 + 32)]/4 = (–1 ± √33)/4
(iii) x = [–4√3 ± √(48 – 48)]/8 = –√3/2, –√3/2
(iv) Discriminant = 1 – 32 = –31 → No real roots
(i) x = [7 ± √(49 – 24)]/4 = [7 ± 5]/4 → 3, 1/2
(ii) x = [–1 ± √(1 + 32)]/4 = (–1 ± √33)/4
(iii) x = [–4√3 ± √(48 – 48)]/8 = –√3/2, –√3/2
(iv) Discriminant = 1 – 32 = –31 → No real roots
33 marks
Find the roots: (i) x – 1/x = 3, x ≠ 0 (ii) 1/(x+4) – 1/(x–7) = 11/30, x ≠ –4,7
Answer: (i) x = (3 ± √13)/2 (ii) x = 2, 1
43 marks
The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age.
Answer: 7 years
53 marks
In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Answer: Maths = 12, English = 18 or Maths = 13, English = 17
63 marks
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Answer: Shorter side = 90 m, Longer side = 120 m
73 marks
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
Answer: Numbers: 18 and 12 or –18 and –12
83 marks
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the original speed.
Answer: 40 km/h
93 marks
Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
Answer: Smaller tap: 25 hours, Larger tap: 15 hours
103 marks
An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.
Answer: Passenger = 33 km/h, Express = 44 km/h
113 marks
Sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.
Answer: Sides: 12 m and 18 m
Exercise 4.4 (Page 91) – Nature of Roots
12 marks each
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x² – 3x + 5 = 0 (ii) 3x² – 4√3 x + 4 = 0 (iii) 2x² – 6x + 3 = 0
(i) 2x² – 3x + 5 = 0 (ii) 3x² – 4√3 x + 4 = 0 (iii) 2x² – 6x + 3 = 0
Answer:
(i) Discriminant = 9 – 40 = –31 → No real roots
(ii) Discriminant = 48 – 48 = 0 → Roots: 2/√3, 2/√3 (equal)
(iii) Discriminant = 36 – 24 = 12 → Roots: (3 ± √3)/2
(i) Discriminant = 9 – 40 = –31 → No real roots
(ii) Discriminant = 48 – 48 = 0 → Roots: 2/√3, 2/√3 (equal)
(iii) Discriminant = 36 – 24 = 12 → Roots: (3 ± √3)/2
23 marks
Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x² + kx + 3 = 0 (ii) kx(x – 2) + 6 = 0
(i) 2x² + kx + 3 = 0 (ii) kx(x – 2) + 6 = 0
Answer: (i) k = ±2√6 (ii) k = 6
33 marks
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.
Answer: Yes, breadth = 20 m, length = 40 m
43 marks
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.
Answer: Not possible (discriminant negative)
Key Concepts Summary
Note 1Standard Form
What is the standard form of a quadratic equation?
Answer: ax² + bx + c = 0, where a ≠ 0, and a, b, c are real numbers.
Note 2Quadratic Formula
What is the quadratic formula for solving ax² + bx + c = 0?
Answer: x = [–b ± √(b² – 4ac)] / 2a
Note 3Nature of Roots
How can we determine the nature of roots using discriminant (D = b² – 4ac)?
Answer:
• D > 0 → Two distinct real roots
• D = 0 → Two equal real roots
• D < 0 → No real roots (imaginary)
• D > 0 → Two distinct real roots
• D = 0 → Two equal real roots
• D < 0 → No real roots (imaginary)
Study materials
- Refernce Books
- NCERT Solutions
- Syllabus
Send Us A Message
Latest posts
News
Navigating the Shift: A Comprehensive Guide to the Revised NCERT Class 9 Syllabus (2026-27)
Read More »
February 25, 2026
No Comments
Blog
January 24, 2026
No Comments
