NCERT Solutions for Class 10 Mathematics Chapter Quadratic Equations

This worksheet provides NCERT Solutions for Class 10 Mathematics Chapter Quadratic Equations. This chapter focuses on understanding quadratic equations, identifying them, and solving them using different algebraic methods. It is an important topic in mathematics that helps students build strong problem-solving skills and prepares them for higher-level concepts. This worksheet includes complete and accurate NCERT solutions strictly based on the given content, helping students practice effectively and understand each step clearly.

Chapter summary: 

This chapter is concept-based. It focuses on mathematical expressions and real-life problem-solving situations. Students work on equations, numerical problems, and application-based questions. The main learning focus is understanding quadratic equations and applying them in different scenarios such as finding numbers, areas, speeds, and ages.

What this NCERT chapter covers?

• Understanding quadratic equations and their standard form 
• Identifying whether a given equation is quadratic or not 
• Solving equations using factorization 
• Working with real-life application problems 
• Using the discriminant to determine the nature of roots 
• Strengthening algebraic manipulation skills 

How to use these NCERT solutions?

• Students should first try solving all questions independently 
• After attempting, they can use these solutions to check accuracy 
• Parents and teachers can guide students by reviewing each step 
• These solutions follow the exact NCERT worksheet order 
• This helps in structured revision and better understanding 

Student tips & learning tricks

• Always simplify equations carefully before solving 
• Check the highest power of the variable to confirm it is quadratic 
• Practice factorization regularly to improve speed 
• Avoid calculation mistakes while expanding expressions 
• Pay attention to signs while solving equations 

Why NCERT solutions are important?

NCERT solutions help students understand the exact answering pattern expected in exams. They build a strong foundation in mathematics and improve accuracy. By following NCERT-aligned answers, students gain confidence and become better prepared for school assessments and exams.
 

Complete answer key – NCERT solutions

Exercise 4.1

1. (i) (x + 1)² = 2(x – 3) 
⇒ x² + 2x + 1 = 2x – 6 
⇒ x² + 2x + 1 – 2x + 6 = 0 
⇒ x² + 7 = 0 
Quadratic equation 

2. (ii) x² – 2x = (–2)(3 – x) 
⇒ x² – 2x = –6 + 2x 
⇒ x² – 4x + 6 = 0 
Quadratic equation 

3. (iii) (x – 2)(x + 1) = (x – 1)(x + 3) 
⇒ x² – x – 2 = x² + 2x – 3 
⇒ –x – 2 – 2x + 3 = 0 
⇒ –3x + 1 = 0 
⇒ 3x – 1 = 0 
Not a quadratic equation 

4. (iv) (x – 3)(2x + 1) = x(x + 5) 
⇒ 2x² – 5x – 3 = x² + 5x 
⇒ x² – 10x – 3 = 0 
Quadratic equation 

5. (v) (2x – 1)(x – 3) = (x + 5)(x – 1) 
⇒ 2x² – 7x + 3 = x² + 4x – 5 
⇒ x² – 11x + 8 = 0 
Quadratic equation 

6. (vi) x² + 3x + 1 = (x – 2)² 
⇒ x² + 3x + 1 = x² – 4x + 4 
⇒ 7x – 3 = 0 
Not a quadratic equation 

7. (vii) (x + 2)³ = 2x(x² – 1) 
⇒ x³ + 6x² + 12x + 8 = 2x³ – 2x 
⇒ –x³ + 6x² + 14x + 8 = 0 
⇒ x³ – 6x² – 14x – 8 = 0 
Not a quadratic equation 

8. (viii) x³ – 4x² – x + 1 = (x – 2)³ 
⇒ x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8 
⇒ 2x² – 13x + 9 = 0 
Quadratic equation 

9. Let first integer = x 
Second integer = x + 1 
⇒ x(x + 1) = 306 
⇒ x² + x – 306 = 0 

10. Let breadth = x 
Length = 2x + 1 
Area = 528 
⇒ x(2x + 1) = 528 
⇒ 2x² + x – 528 = 0 

11. Let Rohan’s age = x 
Mother’s age = x + 26 
After 3 years: 
(x + 3)(x + 29) = 360 
⇒ x² + 32x + 87 = 360 
⇒ x² + 32x – 273 = 0 

12. Let speed = x km/h 
Time = 480/x 
New speed = x – 8 
Time = 480/(x – 8) 
⇒ 480/(x – 8) = 480/x + 3 
⇒ 480x = 480(x – 8) + 3x(x – 8) 
⇒ 480x = 480x – 3840 + 3x² – 24x 
⇒ 3x² – 24x – 3840 = 0 
⇒ x² – 8x – 1280 = 0 

Exercise 4.2

1. (i) x² – 3x – 10 = 0 
⇒ (x – 5)(x + 2) = 0 
x = 5, –2 

2. (ii) 2x² + x – 6 = 0 
⇒ (2x – 3)(x + 2) = 0 
x = 3/2, –2 

3. (iii) 2x² + 7x + 5 = 0 
⇒ (2x + 5)(x + 1) = 0 
x = –5/2, –1 

4. (iv) 2x² – x + 1/8 = 0 
⇒ (4x – 1)(4x – 1) = 0 
x = 1/4, 1/4 

5. (v) 100x² – 20x + 1 = 0 
⇒ (10x – 1)² = 0 
x = 1/10, 1/10 

6. (i) x² – 45x + 324 = 0 
⇒ (x – 9)(x – 36) = 0 
x = 9, 36 

7. (ii) x² – 55x + 750 = 0 
⇒ (x – 25)(x – 30) = 0 
x = 25, 30 

8. Let numbers be x and (27 – x) 
⇒ x(27 – x) = 182 
⇒ x² – 27x + 182 = 0 
⇒ (x – 13)(x – 14) = 0 
Numbers = 13, 14 

9. Let integers be x and x + 1 
⇒ x² + (x + 1)² = 365 
⇒ 2x² + 2x + 1 = 365 
⇒ 2x² + 2x – 364 = 0 
⇒ x² + x – 182 = 0 
⇒ (x – 13)(x + 14) = 0 
x = 13 
Integers = 13, 14 

10. Let base = x 
Altitude = x – 7 
Hypotenuse = 13 
⇒ x² + (x – 7)² = 169 
⇒ 2x² – 14x + 49 = 169 
⇒ 2x² – 14x – 120 = 0 
⇒ x² – 7x – 60 = 0 
⇒ (x – 12)(x + 5) = 0 
x = 12 
Base = 12 cm 
Altitude = 5 cm 

11. Let number of articles = x 
Cost per article = 2x + 3 
⇒ x(2x + 3) = 90 
⇒ 2x² + 3x – 90 = 0 
⇒ (2x + 15)(x – 6) = 0 
x = 6 
Articles = 6 
Cost = 15 

Exercise 4.3

1. (i) 2x² – 3x + 5 = 0 
D = b² – 4ac = 9 – 40 = –31 
No real roots 

2. (ii) 3x² – 4√3 x + 4 = 0 
D = 48 – 48 = 0 
Equal roots 
x = 2/√3 

3. (iii) 2x² – 6x + 3 = 0 
D = 36 – 24 = 12 > 0 
x = (6 ± √12)/4 = (3 ± √3)/2 

4. (i) D = 0 
k² – 24 = 0 
k = ±2√6 

5. (ii) kx(x – 2) + 6 = 0 
⇒ kx² – 2kx + 6 = 0 
D = (–2k)² – 24k = 4k² – 24k = 0 
⇒ 4k(k – 6) = 0 
k = 6 

6. Let length = x, breadth = y 
⇒ 2(x + y) = 80 ⇒ x + y = 40 
⇒ xy = 400 
⇒ x(40 – x) = 400 
⇒ x² – 40x + 400 = 0 
⇒ (x – 20)² = 0 
x = 20 
Length = 20 m 
Breadth = 20 m 

7. Let breadth = x 
Length = 2x 
⇒ 2x² = 800 
⇒ x² = 400 
x = 20 
Breadth = 20 m 
Length = 40 m 

8. Let ages = x and (20 – x) 
⇒ (x – 4)(16 – x) = 48 
⇒ x² – 20x + 64 = 48 
⇒ x² – 20x + 16 = 0 
⇒ (x – 4)(x – 16) = 0 
Ages = 4 and 16 

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