1st Year Math Chapter 3 Exercise 3.2 Notes: Nature of Roots and Relations Between Roots and Coefficients (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 3.2 builds on the solving methods from Exercise 3.1, focusing on what the discriminant reveals about a quadratic equation’s roots, and how the coefficients a, b, c relate directly to the sum and product of those roots — without needing to solve the equation first.
What Does This Exercise Cover?
This exercise teaches how to analyze a quadratic equation’s roots quickly using the discriminant, and how to move between an equation and its roots using the sum and product relationships.
Key Concepts
1. The Discriminant
For ax² + bx + c = 0, the discriminant is D = b² − 4ac. Its value — without solving the equation — determines the nature of the roots.
2. Nature of Roots Based on the Discriminant
The sign of D splits into three cases, shown in the table below.
3. Sum and Product of Roots
For ax² + bx + c = 0 with roots α and β: the sum of the roots is α + β = −b/a, and the product of the roots is αβ = c/a. These relationships hold without ever solving for α and β individually.
4. Forming a Quadratic Equation from Given Roots
If the sum (S) and product (P) of the desired roots are known, the corresponding quadratic equation can be written directly as x² − Sx + P = 0.
Nature of Roots Based on the Discriminant
| Discriminant (D = b² − 4ac) | Nature of Roots |
| D > 0 | Two distinct real roots |
| D = 0 | One repeated real root (equal roots) |
| D < 0 | Two complex conjugate roots |
Solved Examples
Example 1: Determine the nature of the roots of 2x² − 4x + 2 = 0.
Here a = 2, b = −4, c = 2.
D = (−4)² − 4(2)(2) = 16 − 16 = 0.
Answer: D = 0, so the equation has one repeated real root
Example 2: Find the sum and product of the roots of 2x² − 7x + 3 = 0 without solving it.
Here a = 2, b = −7, c = 3.
Sum of roots = −b/a = −(−7)/2 = 7/2.
Product of roots = c/a = 3/2.
Answer: Sum = 7/2, Product = 3/2
Example 3: Form the quadratic equation whose roots are 4 and −2.
Sum: S = 4 + (−2) = 2.
Product: P = 4 × (−2) = −8.
Using x² − Sx + P = 0: x² − 2x + (−8) = 0.
Answer: x² − 2x − 8 = 0
Sample MCQs
1. The discriminant of ax² + bx + c = 0 is given by:
a) b² + 4ac b) b² − 4ac c) −b² − 4ac d) 4ac − b²
Answer: b) b² − 4ac
2. If the discriminant is negative, the roots of the quadratic equation are:
a) Real and equal b) Real and distinct c) Complex conjugates d) Always rational
Answer: c) Complex conjugates
3. For ax² + bx + c = 0 with roots α and β, the sum of roots equals:
a) c/a b) −b/a c) b/a d) −c/a
Answer: b) −b/a
4. For ax² + bx + c = 0 with roots α and β, the product of roots equals:
a) −b/a b) c/a c) −c/a d) b/a
Answer: b) c/a
5. If the discriminant equals zero, the quadratic equation has:
a) Two distinct real roots b) Two complex roots c) One repeated real root d) No roots at all
Answer: c) One repeated real root
Important Short Questions
- Define the discriminant of a quadratic equation.
- What does a negative discriminant indicate about the roots?
- State the formulas for the sum and product of the roots of ax² + bx + c = 0.
- Find the sum and product of the roots of x² − 5x + 6 = 0 without solving it.
- Form a quadratic equation whose roots are 3 and −5.
Important Long Questions
- Determine the nature of the roots of 3x² − 2x + 5 = 0 using the discriminant, and explain what the result means.
- Find the sum and product of the roots of 4x² + 7x − 2 = 0, and verify your answer by solving the equation directly.
- Form the quadratic equation whose roots are (2 + √3) and (2 − √3).
- Explain how the value of the discriminant determines whether the roots of a quadratic equation are real and distinct, real and equal, or complex.
How to Approach This Exercise Effectively
- Calculate the discriminant first in any question about the nature of roots — it answers the question directly without solving the equation.
- Memorize the sum (−b/a) and product (c/a) formulas as a pair — they’re almost always tested together.
- When forming an equation from given roots, always compute S and P first, then substitute directly into x² − Sx + P = 0.
- For roots involving surds like (2 + √3) and (2 − √3), notice the sum eliminates the square root — use this to simplify quickly.
- Cross-check sum/product answers by briefly verifying them against the quadratic formula’s result when time allows.
FAQs
Q: Why is the discriminant useful?
A: It tells you the nature of a quadratic equation’s roots (real and distinct, real and equal, or complex) without needing to solve the equation at all.
Q: Can a quadratic equation have exactly one root?
A: It has exactly one repeated root when the discriminant equals zero — technically this counts as two equal roots, not a single unique root.
Q: How are the sum and product of roots useful in practice?
A: They let you quickly verify solutions, or work backward to construct a quadratic equation when only the roots (not the original equation) are known.
Q: Why do complex roots always come in conjugate pairs here?
A: Since the coefficients a, b, c are real, this follows the same conjugate root rule from Chapter 1 — complex roots of a real-coefficient polynomial always occur in pairs.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 3: Theory of Quadratic Functions, Exercise 3.2.