Is this solution according to PECTAA syllabus?

Yes, these solutions follow the PECTAA / National Curriculum 2023 syllabus for Class 10 Mathematics.

Exercise 2.4 Solution — Chapter 2 Quadratic Equations | Class 10 Math

Q: What is taught in Exercise 2.4 of Class 10 Math Chapter 2?

Exercise 2.4 covers the nature of roots of quadratic equations using the discriminant. It includes examining whether roots are real, equal, rational, irrational, or imaginary, and finding parameter values for specific root conditions.

Q: How many questions are there in Chapter 2 Exercise 2.4?

Exercise 2.4 has 5 questions covering examination of nature of roots (6 parts), finding values of parameters for real and unequal roots, equal roots, and possible values for given conditions.

Q: What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation ax squared plus bx plus c equals 0 is b squared minus 4ac. It determines the nature of roots: greater than 0 (real and unequal), equal to 0 (real and equal), less than 0 (imaginary).

Q: What are the conditions for different types of roots?

For ax squared plus bx plus c equals 0: discriminant greater than 0 gives real and unequal roots (rational if perfect square, irrational if not), discriminant equals 0 gives real and equal roots, and discriminant less than 0 gives imaginary (complex) roots.

Q: How do you find parameter values for equal roots?

To find parameter values for equal roots, set the discriminant equal to 0 and solve for the parameter.

Q: Is this Exercise 2.4 solution valid for all Punjab Boards?

Yes, the content follows the unified Punjab textbook and is applicable to all 10 BISE Punjab boards.

Q: Are solved PDF notes available for Exercise 2.4?

Yes, a complete solved PDF for Exercise 2.4 is embedded on this page and available to download for free.

Q: Is Exercise 2.4 important for Class 10 board exams?

Yes, nature of roots and discriminant are frequently tested in board exams.

Q: Who prepared these Class 10 Math Chapter 2 notes?

These notes were prepared by Muhammad Tayyab, Subject Specialist Mathematics at Govt Christian High School Daska.

Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska

📌 Based on National Curriculum 2023 / PECTAA Syllabus

📖 What's Inside: This exercise covers examination of nature of roots using discriminant (\( \Delta = b^2-4ac \)), finding parameter values for real/equal/imaginary roots. Perfect for Punjab Boards exam preparation.

⬇️ Download PDF (Exercise 2.4 Solved)

📚 Related Resources – Chapter 2: Quadratic Equations

📖 Exercise 2.1 📐 Exercise 2.2 🔢 Exercise 2.3 📈 Exercise 2.5 ⚙️ Exercise 2.6 🧮 Exercise 2.7 🔄 Review Exercise 2

Class 10 Math Chapter 2 Exercise 2.4 – Nature of Roots: Complete Guide

Exercise 2.4 of Chapter 2 introduces the discriminant and its role in determining the nature of roots of quadratic equations. Students learn to analyze whether roots are real, equal, rational, irrational, or imaginary using \( \Delta = b^2 - 4ac \).

What You Will Learn

By working through Exercise 2.4, students master: calculating the discriminant \( \Delta = b^2 - 4ac \), determining the nature of roots based on the discriminant value, finding parameter values for real and unequal roots, and finding parameter values for equal roots.

Topics Covered in This Exercise

Discriminant Analysis – \( \Delta = b^2 - 4ac \)
Real and Unequal Roots – \( \Delta > 0 \) (rational if perfect square, irrational if not)
Real and Equal Roots – \( \Delta = 0 \)
Imaginary Roots – \( \Delta < 0 \)
Parameter Values – finding values for specific root conditions

Why Exercise 2.4 Is Important

Exercise 2.4 is crucial for understanding the nature of roots. Board exam papers frequently feature questions on discriminant analysis and finding parameter values for specific root conditions. This exercise builds essential skills for advanced algebra.

Punjab Board Preparation

Students preparing for board exams under any of the 10 BISE Punjab boards should prioritise this exercise. Discriminant analysis is a frequently tested topic. Practising every part of Q1 and Q2–Q5 by hand is highly recommended.

Exam Tips for Nature of Roots

Remember: \( \Delta = b^2 - 4ac \) for \( ax^2 + bx + c = 0 \)
\( \Delta > 0 \) → real and unequal roots
\( \Delta = 0 \) → real and equal roots
\( \Delta < 0 \) → imaginary (complex) roots
If \( \Delta > 0 \) and is a perfect square → rational roots
If \( \Delta > 0 \) and is not a perfect square → irrational roots

Common Mistakes Students Make

Forgetting to include the negative sign in the discriminant formula.
Not simplifying the discriminant correctly.
Confusing the conditions for rational vs irrational roots.
Making sign errors when solving inequalities.

Muhammad Tayyab Subject Specialist Mathematics

MSc Mathematics · Govt Christian High School Daska, Sialkot, Punjab

Content reviewed against the PECTAA / National Curriculum 2023 syllabus for Class 10 Mathematics, applicable to all 10 BISE Punjab boards.

Last updated: August 7, 2026 Source: Punjab Curriculum & Textbook Board (PCTB)

📖 Exercise 2.4 – Solved Problems

1 Examine the nature of roots of the following quadratic equations:

(i) \(3x^2 - 9x - 2 = 0\) \[ \begin{aligned} a = 3,\; b = -9,\; c = -2 \\ \Delta &= b^2 - 4ac = (-9)^2 - 4(3)(-2) = 81 + 24 = 105 > 0 \end{aligned} \]

As \(\Delta > 0\) and not a perfect square, the roots are irrational and unequal.

(ii) \(x^2 + 6x + 9 = 0\) \[ \begin{aligned} a = 1,\; b = 6,\; c = 9 \\ \Delta &= 36 - 4(1)(9) = 36 - 36 = 0 \end{aligned} \]

As \(\Delta = 0\), the roots are rational (real) and equal.

(iii) \(2x^2 + 4x + 5 = 0\) \[ \begin{aligned} a = 2,\; b = 4,\; c = 5 \\ \Delta &= 16 - 4(2)(5) = 16 - 40 = -24 < 0 \end{aligned} \]

As \(\Delta < 0\), the roots are imaginary and unequal.

(iv) \(7x^2 - 6x - 1 = 0\) \[ \begin{aligned} a = 7,\; b = -6,\; c = -1 \\ \Delta &= 36 - 4(7)(-1) = 36 + 28 = 64 > 0 \end{aligned} \]

As \(\Delta > 0\) and is a perfect square (\(64 = 8^2\)), the roots are rational and unequal.

(v) \(5x^2 - 2x + 10 = 0\) \[ \begin{aligned} a = 5,\; b = -2,\; c = 10 \\ \Delta &= 4 - 4(5)(10) = 4 - 200 = -194 < 0 \end{aligned} \]

As \(\Delta < 0\), the roots are imaginary and unequal.

(vi) \(x^2 - 8x + 16 = 0\) \[ \begin{aligned} a = 1,\; b = -8,\; c = 16 \\ \Delta &= 64 - 4(1)(16) = 64 - 64 = 0 \end{aligned} \]

As \(\Delta = 0\), the roots are rational (real) and equal.

2 For what values of \(t\) the roots of \(3x^2 + x + 9t = 0\) are real and unequal?

\[ \begin{aligned} a = 3,\; b = 1,\; c = 9t \\ \Delta &= b^2 - 4ac = 1 - 4(3)(9t) = 1 - 108t \\ \text{For real and unequal roots: } & \Delta > 0 \\ 1 - 108t &> 0 \implies 1 > 108t \implies t < \frac{1}{108} \end{aligned} \]

✅ \(t < \frac{1}{108}\)

3 If the quadratic equation \(16x^2 + 7px + 49 = 0\) has equal roots, find the values of \(p\).

\[ \begin{aligned} a = 16,\; b = 7p,\; c = 49 \\ \Delta &= (7p)^2 - 4(16)(49) = 49p^2 - 3136 \\ \text{For equal roots: } & \Delta = 0 \\ 49p^2 - 3136 &= 0 \implies 49p^2 = 3136 \\ p^2 &= 64 \implies p = \pm 8 \end{aligned} \]

✅ \(p = 8\) or \(p = -8\)

4 If the quadratic equation \(4u^2 + 8u + q = 0\) has unequal and real roots, find the possible values for \(q\).

\[ \begin{aligned} a = 4,\; b = 8,\; c = q \\ \Delta &= 64 - 4(4)(q) = 64 - 16q \\ \text{For real and unequal roots: } & \Delta > 0 \\ 64 - 16q &> 0 \implies 64 > 16q \implies q < 4 \end{aligned} \]

✅ \(q < 4\)

5 Find the value for \(m\), if the quadratic equation \(mx^2 - 8x + 1 = 0\) has real and equal roots.

\[ \begin{aligned} a = m,\; b = -8,\; c = 1 \\ \Delta &= 64 - 4(m)(1) = 64 - 4m \\ \text{For equal roots: } & \Delta = 0 \\ 64 - 4m &= 0 \implies 4m = 64 \implies m = 16 \end{aligned} \]

✅ \(m = 16\)

📝 Multiple Choice Questions (Chapter 2 Practice)

1. The discriminant of \(2x^2 - 3x + 5 = 0\) is:

A) \(-31\) B) \(31\) C) \(49\) D) \(-49\)

✅ Correct Answer: (A) \(-31\)
\( \Delta = (-3)^2 - 4(2)(5) = 9 - 40 = -31 \)

2. If \(\Delta = 0\), the roots are:

A) Real and unequal B) Real and equal C) Imaginary D) Irrational

✅ Correct Answer: (B) Real and equal
When \(\Delta = 0\), the quadratic equation has real and equal roots.

3. For \(x^2 + 4x + 4 = 0\), the nature of roots is:

A) Real and unequal B) Real and equal C) Imaginary D) Irrational

✅ Correct Answer: (B) Real and equal
\( \Delta = 16 - 4(1)(4) = 0 \), so roots are real and equal.

4. If roots are real and unequal, then:

A) \(\Delta > 0\) B) \(\Delta = 0\) C) \(\Delta < 0\) D) \(\Delta \le 0\)

✅ Correct Answer: (A) \(\Delta > 0\)
Real and unequal roots occur when the discriminant is greater than zero.

5. For \(4x^2 + 4x + 1 = 0\), the discriminant is:

A) \(0\) B) \(16\) C) \(-16\) D) \(4\)

✅ Correct Answer: (A) 0
\( \Delta = 16 - 4(4)(1) = 16 - 16 = 0 \)

📊 Key Concepts – Nature of Roots

Discriminant: \( \Delta = b^2 - 4ac \) for \( ax^2 + bx + c = 0 \)
\( \Delta > 0 \) and perfect square → rational, unequal
\( \Delta > 0 \) and not perfect square → irrational, unequal
\( \Delta = 0 \) → real and equal (rational)
\( \Delta < 0 \) → imaginary (complex) and unequal

❓ Frequently Asked Questions

What is taught in Exercise 2.4 of Class 10 Math Chapter 2?

Exercise 2.4 covers the nature of roots of quadratic equations using the discriminant \( \Delta = b^2 - 4ac \). It includes examining whether roots are real, equal, rational, irrational, or imaginary, and finding parameter values for specific root conditions.

How many questions are there in Chapter 2 Exercise 2.4?

Exercise 2.4 has 5 questions covering examination of nature of roots (6 parts), finding values of parameters for real and unequal roots, equal roots, and possible values for given conditions.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation \( ax^2+bx+c=0 \) is \( \Delta = b^2 - 4ac \). It determines the nature of roots: if \( \Delta > 0 \) (real and unequal), \( \Delta = 0 \) (real and equal), \( \Delta < 0 \) (imaginary).

What are the conditions for different types of roots?

For \( ax^2+bx+c=0 \): \( \Delta > 0 \) gives real and unequal roots (rational if perfect square, irrational if not), \( \Delta = 0 \) gives real and equal roots, and \( \Delta < 0 \) gives imaginary (complex) roots.

How do you find parameter values for equal roots?

To find parameter values for equal roots, set the discriminant \( \Delta = b^2 - 4ac = 0 \) and solve for the parameter.

Is this solution according to the PECTAA syllabus?

Yes, these solutions are prepared according to the PECTAA / National Curriculum 2023 syllabus for Class 10 Mathematics.

Is this Exercise 2.4 solution valid for all Punjab Boards?

Yes, the content follows the unified Punjab textbook and is applicable to students of all 10 BISE Punjab boards.

Are solved PDF notes available for Exercise 2.4?

Yes, a complete solved PDF for Exercise 2.4 is embedded on this page and available to download for free.

Can I download the Chapter 2 Exercise 2.4 solution as a PDF?

Yes, use the Download PDF button on this page to save the complete solved Exercise 2.4 notes to your device.

Is Exercise 2.4 important for Class 10 board exams?

Yes, nature of roots and discriminant are frequently tested in board exams. Exercise 2.4 builds essential skills in analyzing quadratic equations and finding parameter values.

Who prepared these Class 10 Math Chapter 2 notes?

These notes were prepared by Muhammad Tayyab, Subject Specialist Mathematics at Govt Christian High School Daska, for Hira Science Academy.

📘 Exercise 2.1 📘 Exercise 2.2 📘 Exercise 2.3 📘 Exercise 2.5 📘 Exercise 2.6 📘 Exercise 2.7 📘 Review Exercise 2 📗 Chapter 2 Hub – Quadratic Equations 📚 Class 10 Mathematics – All Chapters 🏠 Hira Science Academy – Home