Class 10 Maths Chapter 2 Exercise 2.7 Solutions 2026 – Quadratic Equations Notes PDF

Unit 2: Quadratic Equations and Inequalities | Exercise 2.7 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 2.7 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.

From a Single Answer to a Range of Answers

Every exercise up to now has asked for the exact values of x that make ax² + bx + c equal to zero. Exercise 2.7 asks a different question: for which values of x is ax² + bx + c greater than zero, or less than zero? Instead of one or two specific numbers, the answer becomes a whole range of values — this is a quadratic inequality, and it closes out the chapter by tying every earlier topic together.

Key Concepts Covered in Exercise 2.7

What a Quadratic Inequality Looks Like

A quadratic inequality has the form ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0 (with a ≠ 0), instead of the “= 0” used for equations. Solving it means finding every value of x that satisfies the inequality, not just isolated points.

Finding the Critical Values

The first step is always to solve the related equation ax² + bx + c = 0 — using factorization, the quadratic formula, or any earlier method — to find the critical values (also called boundary points) that divide the number line into regions.

Testing the Regions

The critical values split the number line into up to three intervals. Picking one test value from each interval and substituting it into the original inequality shows whether that entire interval satisfies the inequality or not.

Writing the Final Solution

Once the correct intervals are identified, the solution is written using inequality notation (like 1 < x < 5) or interval notation, keeping in mind whether the boundary points themselves are included (≤, ≥) or excluded (<, >).

Step-by-Step Solved Examples

Q. No. 1: Solve x² − 5x + 6 > 0.

Solve the equation: x² − 5x + 6 = 0  →  (x − 2)(x − 3) = 0  →  x = 2, x = 3

Critical values split the line into: x < 2,  2 < x < 3,  x > 3

Test x = 0 (x<2): (0)²−5(0)+6 = 6 > 0 ✔ true

Test x = 2.5 (middle): (2.5)²−5(2.5)+6 = −0.25, false

Test x = 4 (x>3): (4)²−5(4)+6 = 2 > 0 ✔ true

Solution: x < 2  or  x > 3

Q. No. 2: Solve x² − 4 ≤ 0.

Solve the equation: x² − 4 = 0  →  (x − 2)(x + 2) = 0  →  x = −2, x = 2

Test x = 0 (middle): 0² − 4 = −4, which is ≤ 0 ✔ true

Since the middle region satisfies it, and endpoints are included (≤):

Solution: −2 ≤ x ≤ 2

Q. No. 3: Solve x² + 2x + 5 > 0.

Check the discriminant first: D = 2² − 4(1)(5) = 4 − 20 = −16

Since D < 0, the equation has no real roots, so the expression never crosses zero.

Testing any value (e.g. x=0): 0²+2(0)+5 = 5 > 0, true for all x

Solution: all real numbers (x ∈ ℝ)

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 2.7 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 identifying critical values or the correct solution interval
• Short Questions: Solving simple inequalities that factor cleanly, like Q. No. 1 and 2 above
• Long Questions: Inequalities requiring the discriminant check first, like Q. No. 3, where the sign never changes

Where This Matters Beyond This Exercise

Notice how this exercise uses factorization (2.1), the quadratic formula (2.3), and the discriminant (2.4) all in a single question — quadratic inequalities are really a capstone application of the whole chapter rather than a brand-new topic. Inequalities like these also appear in optimization problems in economics and engineering, wherever a quantity needs to stay above or below a certain threshold.

Common Mistakes Students Make in Exercise 2.7

• Treating the inequality like an equation and stopping at just the critical values instead of testing intervals
• Forgetting to flip the inequality sign when dividing or multiplying both sides by a negative number
• Including or excluding boundary points incorrectly based on whether the inequality is strict (<, >) or not (≤, ≥)
• Not checking the discriminant first when the expression doesn’t factor with real numbers

Why This Exercise Matters for the Board Exam

Because Exercise 2.7 pulls together factorization, the discriminant, and critical-value testing, it’s often used by examiners to check whether a student has genuinely understood the whole chapter rather than just individual techniques in isolation. Presenting the three clear stages — solve, test, conclude — tends to earn full method marks even if a small arithmetic slip affects the final interval.

Quick Links – Chapter 2: Quadratic Equations and Inequalities

Links for sections other than Exercise 2.7 will be activated as they are published on this site.

Download Exercise 2.7 Notes PDF

Click below to view or download the complete Exercise 2.7 notes in PDF format.

[ Download PDF ]      [ View Online ]

(Google Drive / download links to be embedded here.)

Explore More Class 10 Notes

• Class 10 Maths Chapter 1 – Complex Numbers (All Exercises)
• Class 10 Maths Chapter 2 – Full Chapter Notes (All Exercises)
• Class 10 English Notes
• Class 10 Physics Notes
• Class 10 Chemistry Notes
• Class 10 Biology Notes
• Class 10 Computer Science Notes

Join Our Community

Stay updated with new notes, guess papers, and past papers by following us:

• Facebook Page — join thousands of students
• YouTube Channel — video lectures and explanations
• WhatsApp Group — instant updates and doubt-solving

Frequently Asked Questions (FAQs)

Q1. Are these Exercise 2.7 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 complete all earlier exercises before starting 2.7?

Yes — Exercise 2.7 combines factorization or the quadratic formula (to find critical values) with discriminant checks, so being comfortable with Exercises 2.1, 2.3, and 2.4 is important 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

Have a question about any part of Exercise 2.7 or found something missing? Share your comments and questions below — we’d love to help!