Class 10 Maths Chapter 2 Exercise 2.2 Solutions 2026 – Quadratic Equations Notes PDF

Unit 2: Quadratic Equations and Inequalities | Exercise 2.2 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 2.2 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.

When Factorization Isn’t Enough

Factorization works beautifully when a quadratic expression splits neatly into two simple factors — but many quadratic equations simply don’t factor with whole numbers. Exercise 2.2 introduces a second method that works on every quadratic equation without exception: completing the square. The idea is to reshape the equation so that one side becomes a perfect square trinomial, which can then be solved by taking a square root.

Key Concepts Covered in Exercise 2.2

What a Perfect Square Trinomial Looks Like

An expression like x² + 6x + 9 is a perfect square because it factors as (x + 3)². Completing the square is the technique for turning any quadratic expression into this exact shape, even when it doesn’t start out as one.

Completing the Square When a = 1

For x² + bx + c = 0, move c to the other side, then add (b/2)² to both sides. The left side becomes (x + b/2)², which can be solved by taking the square root of both sides.

Completing the Square When a ≠ 1

When the coefficient of x² isn’t 1, every term must first be divided by a before completing the square. This extra division step is where most calculation slips happen, so this exercise gives it dedicated practice.

Taking the Square Root of Both Sides

Once the equation is in the form (x + p)² = q, taking the square root gives x + p = ±√q. The ± sign is essential — forgetting it means losing one of the two roots.

Step-by-Step Solved Examples

Q. No. 1: Solve x² + 6x − 7 = 0 by completing the square.

x² + 6x = 7

Add (6/2)² = 9 to both sides: x² + 6x + 9 = 7 + 9

(x + 3)² = 16

x + 3 = ±4

x = 1  or  x = −7

Q. No. 2: Solve 2x² − 8x + 3 = 0 by completing the square.

Divide every term by 2: x² − 4x + 1.5 = 0

x² − 4x = −1.5

Add (−4/2)² = 4 to both sides: x² − 4x + 4 = −1.5 + 4

(x − 2)² = 2.5

x − 2 = ±√2.5

x = 2 + √2.5  or  x = 2 − √2.5

Q. No. 3: What must be added to x² − 10x to make it a perfect square trinomial?

Coefficient of x is −10, so half of it is −5

(−5)² = 25

Adding 25 gives x² − 10x + 25 = (x − 5)², a perfect square.

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 2.2 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 what value completes the square for a given expression
• Short Questions: Completing the square for equations where a = 1
• Long Questions: Full completing-the-square solutions where a ≠ 1, showing every step

Where Completing the Square Matters Beyond This Exercise

This method isn’t just a backup for when factorization fails — it’s also the exact method used to derive the quadratic formula in the next exercise, so understanding it here makes that derivation much easier to follow. Completing the square is also the standard technique used later for converting equations of circles and parabolas into their standard graphing form.

Common Mistakes Students Make in Exercise 2.2

• Forgetting to divide every term by a first when a ≠ 1
• Adding (b/2)² to only one side of the equation instead of both
• Dropping the ± sign when taking the square root, resulting in only one root
• Making an arithmetic slip when squaring a fraction, such as (b/2)²

Why This Exercise Matters for the Board Exam

Completing the square is tested both directly, as its own question, and indirectly, as the reasoning behind proving the quadratic formula. A solid grasp of this method also makes the nature-of-roots and vertex-form topics that appear later in the chapter far more intuitive, since they all build on the same idea of reshaping an expression into a perfect square.

Quick Links – Chapter 2: Quadratic Equations and Inequalities

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Download Exercise 2.2 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 2.2 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. Why learn completing the square if factorization already works?

Not every quadratic equation factors neatly with whole numbers. Completing the square works on every quadratic equation without exception, and it is also the method used to derive the quadratic formula in the next exercise.

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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