NCERT Solutions Class 10 Mathematics Chapter 2 Quadratic Equations

Mastering NCERT Solutions Class 10 Mathematics for Chapter 2 Quadratic Equations is essential for scoring top marks in CBSE board examinations. Quadratic equations form the cornerstone of Class 10 Algebra, carrying significant weightage in both standard and basic mathematics question papers. This comprehensive guide provides step-by-step NCERT Mathematics Class 10 solutions, detailed derivations, solved textbook exercises, graphical interpretations, and proven exam strategies aligned with the latest CBSE syllabus.

Chapter Overview: Understanding Quadratic Equations

In standard polynomial algebra, an equation of degree 2 is known as a quadratic equation. The standard form of a quadratic equation in one variable x is given by:

ax² + bx + c = 0, where a ≠ 0 and a, b, c ∈ ℝ

Here, a is the coefficient of x² (leading coefficient), b is the coefficient of x (linear coefficient), and c is the constant term. If a = 0, the equation reduces to a linear equation (bx + c = 0); hence the condition a ≠ 0 is mandatory.

Key topics covered in this NCERT Solutions Class 10 Mathematics Chapter include:

• Standard Form & Identification: Checking whether a given algebraic relationship reduces to a quadratic equation.
• Solution by Factorisation: Finding roots by splitting the linear middle term into factors.
• Quadratic Formula (Sridharacharya Formula): Finding exact real roots using algebraic computation.
• Discriminant & Nature of Roots: Analyzing the value of D = b² - 4ac to determine whether roots are real, distinct, equal, or non-real.
• Real-World Applications & Word Problems: Formulating and solving quadratic equations based on geometry, speed-time-distance, age calculations, and work-rate problems.

Important Formulas, Theorems, and Concepts

1. Standard Quadratic Equation & Roots

A real number α is called a root (or solution) of the quadratic equation ax² + bx + c = 0 if and only if:

a(α)² + b(α) + c = 0

A quadratic equation can have at most two real roots.

2. The Quadratic Formula

For any quadratic equation in standard form ax² + bx + c = 0 (with a ≠ 0), the roots are given by:

x = [-b ± √(b² - 4ac)] / (2a)

3. The Discriminant and Nature of Roots

The quantity D = b² - 4ac is called the discriminant because it discriminates between different types of roots:

Exercise Solutions: Step-by-Step NCERT Solutions

Below are comprehensive, step-by-step CBSE NCERT solutions for representative and high-frequency textbook exercise problems.

Exercise 2.1 / 4.1: Checking Quadratic Equations & Problem Formulation

Problem 1: Check whether (x - 2)² + 1 = 2x - 3 is a quadratic equation.

1. Given: (x - 2)² + 1 = 2x - 3
2. Step 1: Expand the left-hand side using the algebraic identity (a - b)² = a² - 2ab + b²:
   LHS = (x² - 4x + 4) + 1 = x² - 4x + 5
3. Step 2: Equate LHS to RHS and shift all terms to the left-hand side:
   x² - 4x + 5 = 2x - 3
   x² - 4x - 2x + 5 + 3 = 0
   x² - 6x + 8 = 0
4. Step 3: Compare with standard form ax² + bx + c = 0. Here, a = 1 ≠ 0, b = -6, and c = 8.
5. Answer: Yes, the given equation is a quadratic equation.

Problem 2: Formulate the equation — 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 of the plot.

1. Given: Area of plot = 528 m².
2. Let: Breadth of the rectangular plot = x metres.
   Length of the rectangular plot = (2x + 1) metres.
3. Formula: Area of rectangle = Length × Breadth
4. Substitution & Simplification:
   x × (2x + 1) = 528
   2x² + x = 528
   2x² + x - 528 = 0
5. Answer: The required quadratic equation is 2x² + x - 528 = 0.

Exercise 2.2 / 4.2: Solving Quadratic Equations by Factorisation

Problem 3: Find the roots of the quadratic equation √2x² + 7x + 5√2 = 0 by factorisation.

1. Given Equation: √2x² + 7x + 5√2 = 0
2. Step 1 (Splitting the middle term): Find two numbers whose product is a × c = √2 × 5√2 = 10 and whose sum is b = 7.
   The two numbers are 5 and 2 (since 5 × 2 = 10 and 5 + 2 = 7).
3. Step 2 (Rewriting the middle term):
   √2x² + 2x + 5x + 5√2 = 0
4. Step 3 (Factor by grouping): Note that 2 = (√2)(√2).
   √2x(x + √2) + 5(x + √2) = 0
   (x + √2)(√2x + 5) = 0
5. Step 4 (Equate each factor to zero):
   x + √2 = 0 ⇒ x = -√2
   √2x + 5 = 0 ⇒ x = -5/√2 (or -5√2 / 2)
6. Answer: The roots of the equation are x = -√2 and x = -5/√2.

Problem 4: Find two consecutive positive integers, sum of whose squares is 365.

1. Let: First positive integer = x, Second consecutive integer = (x + 1).
2. Given Condition: x² + (x + 1)² = 365
3. Step 1: Expand and simplify:
   x² + (x² + 2x + 1) = 365
   2x² + 2x + 1 - 365 = 0
   2x² + 2x - 364 = 0
   Divide entire equation by 2:
   x² + x - 182 = 0
4. Step 2: Factorize by splitting the middle term (product = -182, sum = 1 ⇒ factors are +14 and -13):
   x² + 14x - 13x - 182 = 0
   x(x + 14) - 13(x + 14) = 0
   (x + 14)(x - 13) = 0
5. Step 3: Solve for x:
   x = 13 or x = -14
   Since the problem specifies positive integers, x = -14 is discarded. Hence, x = 13.
6. Consecutive Integer: x + 1 = 13 + 1 = 14.
7. Answer: The required positive integers are 13 and 14.

Exercise 2.3 / 4.3: Quadratic Formula & Nature of Roots

Problem 5: Find the value of k for which the quadratic equation 2x² + kx + 3 = 0 has two equal real roots.

1. Given Equation: 2x² + kx + 3 = 0
2. Step 1: Identify coefficients: a = 2, b = k, c = 3.
3. Step 2 (Condition for equal roots): A quadratic equation has two equal real roots if and only if Discriminant D = 0 (b² - 4ac = 0).
4. Step 3: Substitute the values:
   k² - 4(2)(3) = 0
   k² - 24 = 0
   k² = 24
   k = ±√24 = ±2√6
5. Answer: The required values of k are k = 2√6 and k = -2√6.

Problem 6: A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

1. Given: Speed of boat in still water = 18 km/h, Distance = 24 km.
2. Let: Speed of stream = x km/h (where x < 18).
3. Speed relative to stream:
   Upstream speed = (18 - x) km/h
   Downstream speed = (18 + x) km/h
4. Time taken:
   Time upstream (t₁) = Distance / Speed = 24 / (18 - x)
   Time downstream (t₂) = Distance / Speed = 24 / (18 + x)
5. Condition: t₁ - t₂ = 1
   [24 / (18 - x)] - [24 / (18 + x)] = 1
6. Step-by-step simplification:
   24 × [(18 + x) - (18 - x)] / [(18 - x)(18 + x)] = 1
   24 × (2x) / (324 - x²) = 1
   48x = 324 - x²
   x² + 48x - 324 = 0
7. Solving using Quadratic Formula:
   a = 1, b = 48, c = -324
   D = b² - 4ac = (48)² - 4(1)(-324) = 2304 + 1296 = 3600
   x = [-48 ± √3600] / (2 × 1) = (-48 ± 60) / 2
   x = (-48 + 60) / 2 = 12 / 2 = 6
   or x = (-48 - 60) / 2 = -108 / 2 = -54
8. Conclusion: Speed cannot be negative, so discard x = -54.
9. Answer: The speed of the stream is 6 km/h.

Common Mistakes and Expert Tips for CBSE Board Exams

• Discarding Negative Signs in Quadratic Formula: Remember that the formula begins with -b. If b = -5, then -b = -(-5) = +5. Forgetting this sign change is one of the most frequent arithmetic errors.
• Forgetting Both Signs in Square Roots (±): When solving k² = 24 or using the quadratic formula, always include both positive and negative branches (±). Omitting the negative root loses full marks in value-of-k questions.
• Rejecting Extraneous Roots Without Justification: In word problems dealing with dimensions, time, or speeds, always write a explicit reasoning sentence such as "Since speed/length cannot be negative, we reject x = -54".
• Condition for Real Roots: When a question states that an equation has "real roots", use the condition D ≥ 0 (not just D > 0, because D = 0 also yields real roots).
• Splitting Middle Term Signs: Check that the product of the two split numbers equals a × c (including their algebraic signs).

Board Exam Relevance and Marks Distribution

In the CBSE Class 10 Board Examination, the Algebra unit accounts for 20 marks out of 80 marks. Quadratic Equations typically carries between 5 to 8 marks divided across multiple question types:

• 1-Mark Questions (MCQ / Assertion-Reason): Identifying quadratic equations, evaluating the discriminant, finding roots of simple equations, or checking the nature of roots.
• 2-Mark / 3-Mark Questions (Short Answer): Solving quadratic equations using factorisation or finding unknown coefficients k for equal roots.
• 4-Mark / 5-Mark Questions (Long Answer & Case-Based): Word problems based on speed-time, pipe filling rates, geometric areas, or cost functions.

Frequently Asked Questions (FAQs)

1. Can a quadratic equation have more than two roots?

No. By the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n roots (counting multiplicity). Therefore, a quadratic equation (degree 2) has at most two roots.

2. What happens if the discriminant (D) is less than zero?

If D = b² - 4ac < 0, the square root of a negative number is not a real number. Hence, the quadratic equation has no real roots (its roots are complex numbers, which are studied in Class 11).

3. Which method is best to solve a quadratic equation in exams?

If the question does not specify a method, first check if factorisation by splitting the middle term is easy. If the factors are non-obvious, involve square roots, or contain large numbers, use the Quadratic Formula directly to avoid calculation mistakes.

More NCERT Solutions and Practice

For more NCERT Solutions Mathematics chapter wise guides, downloadable revision notes, and comprehensive mock question papers, explore our full library of NCERT Solutions step by step modules. Test your speed and conceptual understanding by attempting official board-pattern mock papers and previous years' question banks at qptool.theorify.in.