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I. Chapter Summary
This chapter introduces students to quadratic equations, which are polynomial equations of degree two. It explores standard forms, methods of solving (factorization, completing the square, and using the quadratic formula), and the concept of discriminant to determine the nature of roots. Students also learn to formulate and solve real-life problems using quadratic equations.
II. Key Concepts Covered
| Concept | Explanation |
| Quadratic Equation | An equation of the form $ax^2 + bx + c = 0, \quad \text{where } a \ne 0$ |
| Roots/Solutions | The values of x that satisfy the equation. |
| Methods of Solving | |
| 1. Factorization | |
| 2. Completing the Square | |
| 3. Quadratic Formula: | |
| $x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$ | |
| Discriminant (D) | |
| $D = b^2 – 4ac$ | |
| $\text{If } D > 0 \Rightarrow$ Two real & distinct roots | |
| $\text{If } D = 0 \Rightarrow$ Real & equal roots | |
| $\text{If } D < 0 \Rightarrow$ No real roots | |
| Formulating Quadratic Equations | From real-world situations or word problems. |
III. Important Questions
(A) Multiple Choice Questions (1 Mark)
- What is the discriminant of the equation $2x^2 – 4x + 2 = 0$
a) 8
b) 0 ✅ (PYQ 2021)
c) -4
d) 4 - If the roots of a quadratic equation are real and equal, then D is:
a) > 0
b) < 0
c) = 0 ✅
d) Non-existent - The roots of the equation $x^2 + 5x + 6 = 0$ are:
a) 2, 3
b) -2, -3 ✅
c) 1, -6
d) 3, -2 - Which of the following methods can be used to solve any quadratic equation?
a) Factorization
b) Graphical
c) Quadratic Formula ✅
d) Completing the Square
(B) Short Answer Questions (2/3 Marks)
- Find the roots of the quadratic equation $x^2 – 7x + 12 = 0$ by factorization.
- Determine the nature of roots of the equation $3x^2 + 6x + 5 = 0$
- Solve using the quadratic formula: $2x^2 – 4x + 1 = 0$
- The product of two consecutive positive integers is 132. Find the numbers.
(C) Long Answer Questions (5 Marks)
- Solve $5x^2 – 6x – 2 = 0$ by completing the square method.
- A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 48 minutes less for the journey. Find the speed of the train. (PYQ 2019)
- Form a quadratic equation whose roots are reciprocal of the roots of the equation $3x^2 + 7x + 2 = 0$
- If the difference of the squares of two consecutive odd integers is 84, find the numbers.
(D) HOTS (Higher Order Thinking Skills)
- Find the value of k for which the equation $2x^2 + kx + 8 = 0$ has equal roots.
- Show that the equation $x^2 + 2x + 10 = 0$ has no real roots, and interpret this graphically.
IV. Key Formulas/Concepts
| Formula/Concept | Use |
| Quadratic Formula | $x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$ |
| Discriminant | $D = b^2 – 4ac$ to determine the nature of roots |
| Factorization Identity | |
| $(x + a)(x + b) = x^2 + (a + b)x + ab$ | |
| – Use in solving by splitting middle term | |
| Completing the Square | Make perfect square trinomial to isolate x |
V. Deleted Portions (CBSE 2025–2026)
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)
| Unit/Chapter | Estimated Marks | Types of Questions Typically Asked |
| Chapter 4: Quadratic Equations | 6–8 Marks | 1 Long Answer, 2 Short Answer, 1–2 MCQs |
VII. Previous Year Questions (PYQs)
| Year | Marks | Question |
| 2021 | 1 | What is the discriminant of the equation $2x^2 – 4x + 2 = 0 \quad \text{with } D = 0$ |
| 2020 | 3 | Solve using the quadratic formula. |
| 2019 | 5 | Train word problem based on time and speed (reduced time scenario). |
| 2018 | 3 | Determine nature of roots and solve if possible. |
VIII. Real-World Application Examples
| Situation | Application |
| Motion problems | Speed and time problems often result in quadratic equations |
| Area/Geometry | Calculating unknown sides when area is given (e.g., rectangular plots) |
| Business/Profit | Maximizing revenue or minimizing cost models |
| Physics | Projectile motion and path equations |
IX. Student Tips & Strategies for Success
Time Management
- Dedicate 20 minutes/day for solving equations.
- Solve different types: factorization, completing square, quadratic formula.
Exam Preparation
- Create a summary sheet for discriminant values and their implications.
- Practice word problems regularly.
Stress Management
- Use flowcharts to decide which method to apply based on the equation type.
- Visualize quadratic graphs to better understand nature of roots.
X. Career Guidance & Exploration (Class-Specific)
For Classes 9–10
Streams
- Science: Engineering, Physics, Data Science
- Commerce: Finance, Analytics
- Arts: Logical Aptitude, Civil Services
Careers
- Mathematician, Statistician, Architect, Game Developer
Exams
- NTSE, RMO, IMO, CUET Quantitative Aptitude
XI. Important Notes
- Refer to https://ncert.nic.in and https://cbseacademic.nic.in for updates.
- Emphasize understanding of discriminant and graphical interpretations.
- Build flexibility in solving using all three methods.