NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations

What Are Quadratic Equations?

A quadratic equation is a polynomial equation of degree 2. Its standard form is ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0. The condition a is not 0 is critical because if a were 0, the equation would become linear (bx + c = 0), not quadratic. The values of x that satisfy the equation are called its roots or solutions. A quadratic equation always has exactly two roots, which may be real or complex, equal or distinct.

Examples of quadratic equations include 2x^2 + 3x - 5 = 0, x^2 - 4 = 0, and x^2 + 1 = 0. Not every equation that looks quadratic actually is one. For instance, (x + 1)^2 = x^2 + 2x + 3 simplifies to 2x - 2 = 0, which is linear because the x^2 terms cancel out. Exercise 4.1 of the NCERT textbook focuses on this skill of checking whether a given equation is quadratic after simplification. Students must expand both sides, rearrange to one side, and check if the highest power of x is 2.

Methods of Solving Quadratic Equations

The CBSE 2026-27 syllabus covers two methods for solving quadratic equations: factorisation and the quadratic formula.

Factorisation (also called the splitting the middle term method) works by expressing the quadratic as a product of two linear factors. For ax^2 + bx + c = 0, find two numbers p and q such that p + q = b and p x q = a x c. Then split the middle term bx into px + qx and factor by grouping. For example, to solve x^2 - 3x - 10 = 0, we need two numbers with product -10 and sum -3, which are -5 and 2. Splitting: x^2 - 5x + 2x - 10 = x(x-5) + 2(x-5) = (x+2)(x-5) = 0, giving x = -2 or x = 5.

The zero product property is fundamental here: if the product of two expressions is zero, then at least one of them must be zero. This is why we can set each factor equal to zero.

The Quadratic Formula

The quadratic formula provides a universal method to find the roots of any quadratic equation ax^2 + bx + c = 0:

x = [-b +- sqrt(b^2 - 4ac)] / (2a)

This formula is derived by the method of completing the square (which has been removed from the CBSE syllabus but the formula itself is retained). The formula works for every quadratic equation, whether or not it can be factorised easily. For example, 2x^2 - 6x + 3 = 0 cannot be neatly factorised, but using the formula with a = 2, b = -6, c = 3 gives x = [6 +- sqrt(36-24)]/4 = [6 +- 2sqrt(3)]/4 = (3 +- sqrt(3))/2.

Students should memorise this formula thoroughly, as it is used directly in board exam questions and also as a conceptual foundation for understanding the discriminant.

Discriminant and Nature of Roots

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac. It determines the nature of roots without actually solving the equation:

• If D > 0: The equation has two distinct real roots. The roots are x = (-b + sqrt(D))/(2a) and x = (-b - sqrt(D))/(2a). If D is a perfect square, the roots are rational; otherwise, they are irrational.
• If D = 0: The equation has two equal real roots (also called repeated or coincident roots). Both roots equal x = -b/(2a).
• If D < 0: The equation has no real roots. The roots are complex numbers (beyond the scope of Class 10).

This concept is heavily tested in CBSE board exams. The most common question type is: "Find the value of k for which the equation has equal roots." To solve these, set D = 0 and solve for k. Always check that the value of k does not make the leading coefficient zero, as that would make the equation non-quadratic.

For example, if kx(x - 2) + 6 = 0 must have equal roots: kx^2 - 2kx + 6 = 0. D = 4k^2 - 24k = 0, giving k(k-6) = 0. Since k = 0 would eliminate the x^2 term, only k = 6 is valid.

Word Problems

Word problems are a major component of this chapter, contributing 3-4 marks in most board exams. The key is to translate the English statement into a mathematical equation. Common types include:

1. Number problems: Find two consecutive integers whose product is 306. Let them be x and x+1: x(x+1) = 306 gives x^2 + x - 306 = 0.

2. Geometry problems: The altitude of a right triangle is 7 cm less than its base, and the hypotenuse is 13 cm. Using Pythagoras theorem: base^2 + altitude^2 = 13^2 leads to a quadratic equation.

3. Speed-distance-time problems: A train travels 480 km at uniform speed. If speed were 8 km/h less, it would take 3 hours more. Setting up: 480/(x-8) - 480/x = 3 leads to x^2 - 8x - 1280 = 0.

4. Age problems: Rohan's mother is 26 years older. Product of their ages 3 years hence is 360. This gives (x+3)(x+29) = 360, leading to a quadratic in x.

5. Area problems: A rectangular park has perimeter 80 m and area 400 sq m. This leads to x^2 - 40x + 400 = 0, where D = 0 means the park is actually a square.

For each word problem, always define the variable clearly, form the equation, solve it, and then interpret the result in the context of the problem. Reject any root that does not make physical sense (negative length, speed, age, etc.).

How Tested in Board Exams

Quadratic equations is one of the most important chapters for CBSE Class 10, appearing in every board exam with 6-8 marks. The typical question distribution is:

• 1 MCQ (1 mark): Usually on discriminant, nature of roots, or a direct computation like finding sum/product of roots.
• 1 Short Answer (2-3 marks): Solving by factorisation, finding k for equal roots, or finding the discriminant.
• 1 Long Answer/Word Problem (3-4 marks): Speed-distance, area, or age problem that requires forming and solving a quadratic equation.

Since CBSE 2023, case study questions involving quadratic equations have become common. These present a real-life scenario (swimming pool dimensions, train journey) and ask 4 sub-questions worth 1+1+2 marks. The questions test equation formation, solving by factorisation, and discriminant computation.

Assertion-reason questions test conceptual understanding of discriminant conditions and the definition of quadratic equations. Common traps include: D = 0 means "one root" (wrong - it means two equal roots), and forgetting to reject k = 0.

Tips for Full Marks

1. Always write the standard form ax^2 + bx + c = 0 before applying any formula. Identify a, b, and c explicitly.
2. For factorisation questions, show all steps: splitting, grouping, factoring, and stating both roots.
3. For word problems, the first line should define the variable: "Let the speed = x km/h." The last line should state the answer in words: "Therefore, the speed of the train is 40 km/h."
4. For discriminant questions, write D = b^2 - 4ac with values substituted, then simplify step by step.
5. For "find k" questions, clearly state the condition (D = 0 for equal roots, D > 0 for real roots, etc.) before solving.
6. Verify your answers: substitute roots back into the original equation, and check that word problem answers make physical sense.

Connection to Other Chapters

Quadratic equations connect to several other chapters in Class 10 Mathematics:

• Chapter 2 (Polynomials): The roots of ax^2 + bx + c = 0 are the zeroes of the polynomial p(x) = ax^2 + bx + c. The relationships sum of roots = -b/a and product of roots = c/a are directly from the polynomial chapter.
• Chapter 3 (Pair of Linear Equations): Some problems can be reduced to quadratic equations after substitution. Understanding both linear and quadratic solving methods gives a complete toolkit.
• Chapter 5 (Arithmetic Progressions): Problems like "the sum of first n terms equals a certain value" can lead to quadratic equations in n.
• Chapter 6 (Triangles) and Chapter 8 (Trigonometry): Geometry problems involving Pythagoras theorem or trigonometric identities sometimes reduce to quadratic equations.

Mastering this chapter builds algebraic fluency that is essential for Class 11 and 12 Mathematics, where quadratic expressions appear in coordinate geometry, calculus, and complex numbers.