Class 10 Maths Chapter 2 Exercise 2.4 Solutions 2026 – Quadratic Equations Notes PDF

Unit 2: Quadratic Equations and Inequalities | Exercise 2.4 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 2.4 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.

Predicting the Roots Before Solving

Exercise 2.3 showed that the expression under the square root in the quadratic formula, b² − 4ac, decides whether the final answer involves a real number or an imaginary one. Exercise 2.4 studies this expression on its own, under the name discriminant, and shows that its value alone — without solving the equation at all — is enough to predict exactly what kind of roots an equation will have.

Key Concepts Covered in Exercise 2.4

The Discriminant

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. It is simply the part of the quadratic formula that sits inside the square root, isolated and studied as its own quantity.

Two Real, Distinct Roots (D > 0)

When the discriminant is positive, its square root is a real number, so the formula produces two different real values of x.

Two Equal Real Roots (D = 0)

When the discriminant is exactly zero, the ± in the formula makes no difference — both roots come out identical, giving one repeated real root.

No Real Roots (D < 0)

When the discriminant is negative, its square root is imaginary, so the two roots are a complex conjugate pair rather than real numbers — the same situation seen in Exercise 2.3’s complex-root example.

Step-by-Step Solved Examples

Q. No. 1: Without solving, determine the nature of the roots of x² − 4x + 4 = 0.

a = 1, b = −4, c = 4

D = b² − 4ac = (−4)² − 4(1)(4) = 16 − 16 = 0

Since D = 0, the equation has two equal real roots.

Q. No. 2: Without solving, determine the nature of the roots of 2x² + 3x + 5 = 0.

a = 2, b = 3, c = 5

D = b² − 4ac = 3² − 4(2)(5) = 9 − 40 = −31

Since D < 0, the equation has no real roots (the roots are complex).

Q. No. 3: Find the value of k for which x² + kx + 9 = 0 has two equal real roots.

For equal roots, D = 0: k² − 4(1)(9) = 0

k² − 36 = 0

k² = 36

k = 6  or  k = −6

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 2.4 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 which root case a given discriminant value belongs to
• Short Questions: Calculating the discriminant and stating the nature of roots without fully solving the equation
• Long Questions: Finding an unknown constant (like k above) that produces equal, real, or complex roots

Where the Discriminant Matters Beyond This Exercise

Knowing the discriminant lets a student check, in seconds, whether an equation is even worth solving for real answers — useful in word problems where only real, physically meaningful solutions make sense (a length or a time, for instance, can never be complex). This same b² − 4ac test reappears later in the chapter when analyzing quadratic inequalities.

Common Mistakes Students Make in Exercise 2.4

• Substituting b² as b × 2 instead of b × b
• Forgetting the negative sign in −4ac when c itself is negative
• Confusing D = 0 (equal roots) with D < 0 (no real roots) — these are easy to mix up
• Solving the whole equation with the quadratic formula when the question only asks for the nature of the roots

Why This Exercise Matters for the Board Exam

Discriminant questions are fast to answer once the formula is memorized, since they require only substitution and a sign check — no further solving needed. This makes Exercise 2.4 one of the highest mark-per-minute topics in the chapter, and finding unknown constants (as in Q. No. 3) is a favorite long-question style across Punjab boards.

Quick Links – Chapter 2: Quadratic Equations and Inequalities

Links for sections other than Exercise 2.4 will be activated as they are published on this site.

Download Exercise 2.4 Notes PDF

Click below to view or download the complete Exercise 2.4 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 2.4 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 fully solve an equation to find the nature of its roots?

No — that’s the whole point of the discriminant. Calculating just b² − 4ac and checking its sign is enough to determine the nature of the roots without solving for x 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

Have a question about any part of Exercise 2.4 or found something missing? Share your comments and questions below — we’d love to help!