Class 10 Maths Chapter 2 Exercise 2.5 Solutions 2026 – Quadratic Equations Notes PDF

Unit 2: Quadratic Equations and Inequalities | Exercise 2.5 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 2.5 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.

What the Coefficients Already Know

Every quadratic equation’s roots are hiding inside its coefficients, long before anyone solves for them. Exercise 2.5 reveals two simple shortcuts: the sum of the roots and the product of the roots can both be read directly from a, b, and c — no factorization, no formula, no square roots required. This turns a lot of previously tedious work into quick substitution.

Key Concepts Covered in Exercise 2.5

Sum of the Roots

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a. This comes directly from adding the two results of the quadratic formula together, since the √(b²−4ac) terms cancel out.

Product of the Roots

Similarly, α × β = c/a. Multiplying the two quadratic-formula results together makes the square root terms disappear through difference of squares, leaving just this clean ratio.

Using Sum and Product Without Finding the Roots

Once α + β and αβ are known, many expressions involving α and β can be evaluated directly — for example, α² + β² = (α + β)² − 2αβ — without ever solving the original equation for its individual roots.

Verifying the Relationships

This exercise also includes questions that go the other way: solve an equation, find its actual roots, then verify that −b/a and c/a match what was calculated directly. This builds confidence that the shortcut formulas are trustworthy.

Step-by-Step Solved Examples

Q. No. 1: Without solving, find the sum and product of the roots of 3x² − 7x + 2 = 0.

a = 3, b = −7, c = 2

Sum of roots = −b/a = −(−7)/3 = 7/3

Product of roots = c/a = 2/3

Q. No. 2: If α and β are the roots of x² − 6x + 4 = 0, find α² + β² without finding α and β individually.

a = 1, b = −6, c = 4

α + β = −b/a = 6, αβ = c/a = 4

α² + β² = (α + β)² − 2αβ = 6² − 2(4) = 36 − 8 = 28

Q. No. 3: Solve x² − 5x + 6 = 0 and verify the sum and product of its roots.

Factorizing: (x − 2)(x − 3) = 0, so α = 2, β = 3

Actual sum: 2 + 3 = 5. Formula: −b/a = −(−5)/1 = 5 ✔

Actual product: 2 × 3 = 6. Formula: c/a = 6/1 = 6 ✔

Both match, verifying the relationships.

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 2.5 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 on stating the sum or product of roots directly from a, b, and c
• Short Questions: Using the sum and product to evaluate simple symmetric expressions like α² + β²
• Long Questions: Verifying the relationships by fully solving an equation and comparing both methods

Where This Matters Beyond This Exercise

This relationship is the foundation for the very next exercise, where it’s used in reverse — building a brand-new quadratic equation just from knowing what its sum and product of roots should be. It also lets students sanity-check an answer quickly: after solving an equation by factorization or the formula, adding and multiplying the two roots should always match −b/a and c/a.

Common Mistakes Students Make in Exercise 2.5

• Forgetting the negative sign in −b/a for the sum of the roots
• Mixing up which formula is for the sum and which is for the product
• Trying to expand (α + β)² incorrectly, forgetting the middle term 2αβ
• Not simplifying a, b, and c to their correct values when the equation isn’t already in standard form

Why This Exercise Matters for the Board Exam

Sum-and-product questions are quick, formula-based, and don’t require solving the full equation, making them efficient marks in the short-question section. They’re also a common way examiners test whether students truly understand the structure of a quadratic equation, rather than just memorizing how to solve one.

Quick Links – Chapter 2: Quadratic Equations and Inequalities

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 2.5 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 to solve the equation to use these formulas?

No — that’s the main advantage of this exercise. The sum (−b/a) and product (c/a) of the roots can be found directly from the coefficients without solving the equation at all.

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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