TEACHGYAN • CLASS 10 MATHEMATICS • NCERT 2026–27

Chapter 4: Quadratic Equations

Detailed Notes • Worked Examples • Competency • Board Practice

📖 Chapter Explanation

Quadratic equations model relationships involving a squared variable. The chapter develops factorisation and the quadratic formula, while the discriminant gives a quick way to determine the nature of roots.

🧭 Concept Journey

standard form → factorisation → quadratic formula → discriminant → nature of roots → applications

📌 Core Concepts

For ax²+bx+c=0, x=[−b±√(b²−4ac)]/(2a). The discriminant D=b²−4ac: D>0 gives two distinct real roots, D=0 gives equal real roots, and D<0 gives no real roots.

💡 Worked Example

Different methods can solve the same equation. Factorisation is efficient when factors are easy to identify; the quadratic formula is general and works whenever a≠0.

🔍 Understanding the Idea

For ax²+bx+c=0, x=[−b±√(b²−4ac)]/(2a). The discriminant D=b²−4ac: D>0 gives two distinct real roots, D=0 gives equal real roots, and D<0 gives no real roots.

⚠️ Common Errors

Check that the equation is actually quadratic and written in standard form. Do not lose the ± sign in the quadratic formula.

🧠 Concept-Based MCQs

The discriminant of x²−4x+4 is: A) −4 B) 0 C) 4 D) 8
A quadratic with D<0 has: A) two distinct real roots B) one real root C) no real roots D) three roots
The quadratic formula contains: A) −b only B) ±√D C) c/a only D) a+b
For x²−5x+6=0, the roots are: A) 1,6 B) 2,3 C) −2,−3 D) 3,5

Answer Key: 1-B, 2-C, 3-B, 4-B.

📊 Case-Based / Competency

A rectangular garden has area 48 m² and its length is 2 m more than its breadth. Form the quadratic equation, solve it and reject any dimension that is not meaningful.

Identify the variables or mathematical structure.
Form the required mathematical representation.
Solve systematically.
Interpret and verify the result.

🔎 Extension Case

A farmer designs a rectangular enclosure with a fixed area and a length condition. Students translate the geometry into a quadratic equation, solve it, interpret both roots and explain why a negative dimension is rejected.

🔥 HOTS & Board Challenge

Explain why the chosen method works.
Create a similar problem with different values and solve it.
Find and correct a plausible incorrect solution.
Connect the concept to a real-life situation.

✍️ Exam Practice

Define the central concept with an example.
Solve a direct application problem with complete steps.
Solve a multi-step competency problem.
Interpret a graph/table/context where applicable.
Write a 5-mark answer with reasoning and verification.

📄 Practice Worksheet

5 concept checks
5 standard problems
3 application problems
2 case-based problems
2 HOTS problems
1 board-style challenge

🎯 One-Minute Revision