NCERT Solutions for Class 10 Mathematics Chapter 4
NCERT solutions for Class 10 Mathematics Chapter 4 Quadratic Equations – complete answers & explanations
Quadratic equations are an important part of Class 10 Mathematics and help students understand how to solve equations involving squares of variables. In this chapter, students learn how to form quadratic equations, solve them using different methods, and apply them to real-life situations like finding ages, dimensions, and speed. This topic builds a strong base for higher mathematics and is very important for board exams. This blog provides clear and reliable NCERT solutions that help students understand the correct approach to answering questions step by step. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance and strengthen your understanding with the right support.
What this NCERT chapter covers?
1. Understanding what quadratic equations are
and how to identify them correctly.
2. Learning how to convert given expressions into standard quadratic form.
3. Solving equations using factorisation methods step by step.
4. Applying quadratic equations to real-life word problems like ages, area, and speed.
5. Understanding the concept of discriminant and how it determines the nature of roots.
6. Learning how to find real, equal, or imaginary roots using formulas.
7. Practicing algebraic manipulation to simplify and solve equations.
8. Developing problem-solving skills through application-based questions.
9. Improving logical reasoning by forming equations from given conditions.
10. Strengthening exam preparation with structured and stepwise solutions.
How to use these NCERT solutions?
1. Start by attempting each question
on your own before checking the answers.
2. Carefully compare your solution with the given answers to identify mistakes.
3. Focus on understanding each step rather than just memorising the answer.
4. Follow the exact order of questions as given to stay aligned with your practice.
5. Use the solutions to learn correct equation formation and solving methods.
6. Parents and teachers can guide students by explaining steps wherever confusion arises.
7. Revise difficult questions multiple times to build confidence.
8. Practice similar questions to strengthen your understanding of concepts.
Important tips & tricks for students
1. Always write equations in standard form before solving them.
2. Check calculations carefully to avoid small mistakes.
3. Learn factorisation techniques properly as they are widely used.
4. Pay attention to word problems and convert them into equations correctly.
5. Understand the discriminant formula clearly to determine roots.
6. Avoid skipping steps as stepwise solutions help in scoring full marks.
7. Practice regularly to improve speed and accuracy.
8. Recheck answers to ensure correctness, especially in board exams.
NCERT solutions – complete answer key
Exercise No. 4.1
(i) (x + 1)² = 2(x – 3)
⇒ x² + 2x + 1 = 2x – 6
⇒ x² + 2x + 1 – 2x + 6 = 0
⇒ x² + 7 = 0
Quadratic equation
(ii) x² – 2x = (–2)(3 – x)
⇒ x² – 2x = –6 + 2x
⇒ x² – 4x + 6 = 0
Quadratic equation
(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
(iv) (x – 3)(2x + 1) = x(x + 5)
⇒ 2x² – 5x – 3 = x² + 5x
⇒ x² – 10x – 3 = 0
Quadratic equation
(v) (2x – 1)(x – 3) = (x + 5)(x – 1)
⇒ 2x² – 7x + 3 = x² + 4x – 5
⇒ x² – 11x + 8 = 0
Quadratic equation
(vi) x² + 3x + 1 = (x – 2)²
⇒ x² + 3x + 1 = x² – 4x + 4
⇒ 7x – 3 = 0
Not a quadratic equation
(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
(viii) x³ – 4x² – x + 1 = (x – 2)³
⇒ x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8
⇒ 2x² – 13x + 9 = 0
Quadratic equation
(i) Let breadth = x
Length = 2x + 1
Area = 528
⇒ x(2x + 1) = 528
⇒ 2x² + x – 528 = 0
(ii) Let first integer = x
Second integer = 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 + 87 = 360
⇒ x² + 32x – 273 = 0
(iv) 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 No. 4.2
(i) x² – 3x – 10 = 0
⇒ (x – 5)(x + 2) = 0
x = 5, –2
(ii) 2x² + x – 6 = 0
⇒ (2x – 3)(x + 2) = 0
x = 3/2, –2
(iii) 2x² + 7x + 5 = 0
⇒ (2x + 5)(x + 1) = 0
x = –5/2, –1
(iv) 2x² – x + 1/8 = 0
⇒ (4x – 1)(4x – 1) = 0
x = 1/4, 1/4
(v) 100x² – 20x + 1 = 0
⇒ (10x – 1)² = 0
x = 1/10, 1/10
2 (i) x² – 45x + 324 = 0
⇒ (x – 9)(x – 36) = 0
x = 9, 36
(ii) x² – 55x + 750 = 0
⇒ (x – 25)(x – 30) = 0
x = 25, 30
3.Let numbers be x and (27 – x)
⇒ x(27 – x) = 182
⇒ x² – 27x + 182 = 0
⇒ (x – 13)(x – 14) = 0
Numbers = 13, 14
4 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
5 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
6 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 No. 4.3
(i) 2x² – 3x + 5 = 0
D = b² – 4ac = 9 – 40 = –31
No real roots
(ii) 3x² – 4√3 x + 4 = 0
D = 48 – 48 = 0
Equal roots
x = 2/√3
(iii) 2x² – 6x + 3 = 0
D = 36 – 24 = 12 > 0
x = (6 ± √12)/4 = (3 ± √3)/2
(i) D = 0
k² – 24 = 0
k = ±2√6
(ii) kx(x – 2) + 6 = 0
⇒ kx² – 2kx + 6 = 0
D = (–2k)² – 24k = 4k² – 24k = 0
⇒ 4k(k – 6) = 0
k = 6
Let length = x m
Breadth = y m
Perimeter = 80
2(x + y) = 80
x + y = 40
y = 40 – x
Area = 400
x(40 – x) = 400
x² – 40x + 400 = 0
(x – 20)² = 0
x = 20
Length = 20 m
Breadth = 20 m
Let breadth = x m
Length = 2x m
Area = 800
2x² = 800
x² = 400
x = 20 (reject negative value)
Breadth = 20 m
Length = 40 m
Let present ages = x and (20 – x)
Four years ago:
(x – 4)(16 – x) = 48
x² – 20x + 112 = 48
x² – 20x + 64 = 0
(x – 16)(x – 4) = 0
x = 16 or 4
Present ages = 16 years and 4 years
Why NCERT solutions help students?
NCERT solutions help students build a strong understanding of concepts, improve problem-solving skills, and prepare effectively for exams. They provide the correct approach to answering questions and ensure that students follow accurate methods, which boosts confidence and helps in scoring better marks.
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