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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)

1. What is the discriminant of the equation $2x^2 – 4x + 2 = 0$
   a) 8
   b) 0 ✅ (PYQ 2021)
   c) -4
   d) 4
2. If the roots of a quadratic equation are real and equal, then D is:
   a) > 0
   b) < 0
   c) = 0 ✅
   d) Non-existent
3. The roots of the equation $x^2 + 5x + 6 = 0$ are:
   a) 2, 3
   b) -2, -3 ✅
   c) 1, -6
   d) 3, -2
4. 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)

1. Find the roots of the quadratic equation $x^2 – 7x + 12 = 0$ by factorization.
2. Determine the nature of roots of the equation $3x^2 + 6x + 5 = 0$
3. Solve using the quadratic formula: $2x^2 – 4x + 1 = 0$
4. The product of two consecutive positive integers is 132. Find the numbers.

(C) Long Answer Questions (5 Marks)

1. Solve $5x^2 – 6x – 2 = 0$ by completing the square method.
2. 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)
3. Form a quadratic equation whose roots are reciprocal of the roots of the equation $3x^2 + 7x + 2 = 0$
4. If the difference of the squares of two consecutive odd integers is 84, find the numbers.

(D) HOTS (Higher Order Thinking Skills)

1. Find the value of k for which the equation $2x^2 + kx + 8 = 0$ has equal roots.
2. 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.