Class 10 Maths Chapter 2 Exercise 2.6 Solutions 2026 – Quadratic Equations Notes PDF
Unit 2: Quadratic Equations and Inequalities | Exercise 2.6 | Punjab Board New Syllabus 2026–27
Updated August 2026: Exercise 2.6 solutions for Quadratic Equations have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.
Working Backward This Time
Every exercise so far has started with an equation and asked for its roots. Exercise 2.6 flips that process entirely: it starts with the roots — or clues about them — and asks students to build the original quadratic equation. This is the direct reverse application of the sum-and-product relationship learned in Exercise 2.5.
Key Concepts Covered in Exercise 2.6
The Equation-from-Roots Formula
If α and β are the roots of a quadratic equation, the equation itself can always be written as x² − (α + β)x + αβ = 0 — in words, x² minus the sum of the roots times x, plus the product of the roots, equals zero.
Forming Equations from Given Numerical Roots
When α and β are given directly as numbers, this exercise gives practice simply computing their sum and product, then substituting into the formula above.
Forming Equations from Roots Described Indirectly
Some questions describe the roots without stating them outright — for example, “roots that are reciprocals of the roots of another equation,” or “roots each 2 more than the roots of another equation.” These require finding the new sum and product from the original equation’s sum and product first.
Checking the Result
Once a new equation is formed, it’s good practice to verify it by checking that its own −b/a and c/a match the sum and product that were used to build it — the same verification idea introduced in Exercise 2.5.
Step-by-Step Solved Examples
Q. No. 1: Form the quadratic equation whose roots are 4 and −3.
Sum of roots = 4 + (−3) = 1
Product of roots = 4 × (−3) = −12
Equation: x² − (1)x + (−12) = 0
x² − x − 12 = 0
Q. No. 2: Form the quadratic equation whose roots are 2 + √3 and 2 − √3.
Sum of roots = (2 + √3) + (2 − √3) = 4
Product of roots = (2 + √3)(2 − √3) = 2² − (√3)² = 4 − 3 = 1
Equation: x² − 4x + 1 = 0
Q. No. 3: If α and β are the roots of x² − 3x + 2 = 0, form a new equation whose roots are α + 1 and β + 1.
From the original equation: α + β = 3, αβ = 2
New sum = (α + 1) + (β + 1) = (α + β) + 2 = 3 + 2 = 5
New product = (α + 1)(β + 1) = αβ + α + β + 1 = 2 + 3 + 1 = 6
New equation: x² − 5x + 6 = 0
These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 2.6 in the same numbered, step-by-step format.
MCQs, Short Questions & Long Questions from This Exercise
Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:
- MCQs: Quick questions forming an equation from two simple given roots
- Short Questions: Forming equations from roots given as surds (like Q. No. 2 above)
- Long Questions: Forming a new equation from roots described in terms of another equation’s roots (like Q. No. 3 above)
Where This Matters Beyond This Exercise
This “working backward” skill is common in board exam word problems where a scenario describes a relationship between two unknowns rather than giving an equation directly — students must construct the equation themselves before they can even begin solving it. It also reinforces, from the opposite direction, just how completely a, b, and c determine everything about a quadratic equation’s roots.
Common Mistakes Students Make in Exercise 2.6
- Writing the equation as x² + (sum)x + (product) = 0, forgetting the minus sign before the sum
- Making an error when multiplying surd roots, such as forgetting (√3)² = 3
- In indirect problems, using the new roots’ individual values rather than expressing them in terms of the original sum and product
- Forgetting to expand (α + 1)(β + 1) fully before substituting known values
Why This Exercise Matters for the Board Exam
Forming equations from roots is a compact, reliable long-question type, especially the indirect versions like Q. No. 3, which test genuine understanding of how sum and product transform rather than rote formula use. It’s a strong indicator, for examiners, of whether a student understands the theory behind quadratic equations or has only memorized how to solve them.
Quick Links – Chapter 2: Quadratic Equations and Inequalities
| Section | Covers |
| Exercise 2.1 | Standard form & factorization method |
| Exercise 2.2 | Completing the square |
| Exercise 2.3 | Quadratic formula |
| Exercise 2.4 | Discriminant & nature of roots |
| Exercise 2.5 | Sum & product of roots |
| Exercise 2.6 | You are here |
| Exercise 2.7 | Coming soon |
| Chapter 2 MCQs | Coming soon |
Links for sections other than Exercise 2.6 will be activated as they are published on this site.
Download Exercise 2.6 Notes PDF
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Frequently Asked Questions (FAQs)
Q1. Are these Exercise 2.6 notes free to download?
Yes, all notes on this page are completely free to view and download in PDF format.
Q2. Which board are these notes for?
These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).
Q3. Do I need Exercise 2.5 before starting this one?
Yes — Exercise 2.6 directly reverses the sum-and-product relationship taught in Exercise 2.5, so being comfortable with −b/a and c/a is essential before starting here.
Q4. Does this page include MCQs and short questions too?
Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.
Q5. How can I download the PDF?
Click the “Download PDF” button above and the notes will open or download directly to your device.
Q6. Are these notes updated for the new 2026–27 syllabus?
Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.
Comments & Feedback
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