Class 10 Maths Chapter 2 Exercise 2.3 Solutions 2026 – Quadratic Equations Notes PDF

Unit 2: Quadratic Equations and Inequalities | Exercise 2.3 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 2.3 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.

One Formula for Every Quadratic Equation

Factorization needs the right numbers to line up, and completing the square takes several careful steps. Exercise 2.3 offers a shortcut: apply completing the square once, in general, to ax² + bx + c = 0 itself, and the result is a ready-made formula that solves any quadratic equation by simply substituting a, b, and c. This is the quadratic formula, and it works every single time — even when the roots are irrational or involve complex numbers.

Key Concepts Covered in Exercise 2.3

The Quadratic Formula

For ax² + bx + c = 0 (a ≠ 0), the roots are given by x = (−b ± √(b² − 4ac)) / 2a. This single formula replaces the entire completing-the-square process with one substitution.

Identifying a, b, and c Correctly

Before substituting into the formula, the equation must be in standard form ax² + bx + c = 0. This exercise gives practice correctly reading off a, b, and c, including their signs, from equations that aren’t already tidily arranged.

Substituting Carefully

Most errors in this exercise come from arithmetic, not concept — particularly with negative values of b and c. Writing out −b, b², and −4ac as separate small steps before combining them helps avoid sign mistakes.

Simplifying the Final Answer

After substituting, the expression under the square root (b² − 4ac) must be simplified first, and the final fraction reduced to lowest terms where possible, before the two roots are stated separately.

Step-by-Step Solved Examples

Q. No. 1: Solve x² − 5x + 6 = 0 using the quadratic formula.

a = 1, b = −5, c = 6

x = (−(−5) ± √((−5)² − 4(1)(6))) / 2(1)

x = (5 ± √(25 − 24)) / 2

x = (5 ± √1) / 2 = (5 ± 1) / 2

x = 3  or  x = 2

Q. No. 2: Solve 3x² + 2x − 1 = 0 using the quadratic formula.

a = 3, b = 2, c = −1

x = (−2 ± √(2² − 4(3)(−1))) / 2(3)

x = (−2 ± √(4 + 12)) / 6

x = (−2 ± √16) / 6 = (−2 ± 4) / 6

x = 1/3  or  x = −1

Q. No. 3: Solve x² + 4x + 5 = 0 using the quadratic formula.

a = 1, b = 4, c = 5

x = (−4 ± √(4² − 4(1)(5))) / 2(1)

x = (−4 ± √(16 − 20)) / 2 = (−4 ± √(−4)) / 2

√(−4) = 2i, so x = (−4 ± 2i) / 2 = −2 ± i

The roots are complex: x = −2 + i  or  x = −2 − i

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 2.3 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 correctly identifying a, b, and c from an equation
• Short Questions: Direct substitution into the quadratic formula for equations with real, rational roots
• Long Questions: Equations producing irrational or complex roots, requiring careful simplification of the square root term

Where the Quadratic Formula Matters Beyond This Exercise

Notice that Q. No. 3 above produced roots involving i — this is exactly where Chapter 1 (Complex Numbers) connects back into this chapter, since the quadratic formula is often the first place complex numbers appear as an answer rather than an abstract topic. The expression under the square root, b² − 4ac, also has its own name and importance: it is called the discriminant, and it is studied in detail in the very next exercise.

Common Mistakes Students Make in Exercise 2.3

• Writing −b as b when b is already negative, effectively flipping its sign twice
• Forgetting the 2a in the denominator applies to the entire numerator, not just the square root term
• Making an arithmetic error inside b² − 4ac, especially with negative c
• Stopping after finding the value under the square root instead of simplifying it fully

Why This Exercise Matters for the Board Exam

The quadratic formula is arguably the single most reliable method in the entire chapter, since it works on every quadratic equation without needing to check whether it factors nicely first. For this reason, it’s often the safest method to fall back on in a board exam when time is short or an equation looks unfamiliar.

Quick Links – Chapter 2: Quadratic Equations and Inequalities

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 2.3 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. Should I use the quadratic formula for every equation, even ones that factor easily?

Factorization is usually faster when an equation clearly factors with small whole numbers. The quadratic formula is best used when factorization isn’t obvious, or as a reliable fallback method under exam time pressure.

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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