Chapter 3 of 8

38%

Tamil Nadu (TNBSE)Class 10 Mathematics★ 10-15 marks in the TN SSLC Class 10 Maths public exam through MCQs, short answers and quadratic/matrix problems

Algebra

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 3, Algebra. Simultaneous linear equations in three variables, GCD and LCM of polynomials, rational expressions, square root of polynomials, quadratic equations (solving, discriminant, roots) and matrices. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 24 min readHard difficulty✓ Free · NCERT-aligned

Worksheet

Podcast

Listen as a conversation

A teacher & a student talk through the chapter — in your language.

Mind Map

The chapter at a glance

Every concept and exam-favourite branch — revisable in 5 minutes.

…

Study Guide

Night-before notes

Key points, glossary and the questions most likely to show up.

…

VocabularyBuild your vocabularyLexiGenome mines the key words and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Solve simultaneous linear equations in three variables
2Find the GCD and LCM of polynomials
3Solve quadratic equations and analyse their roots
4Use the discriminant and root relations
5Perform matrix operations

💡

Why this chapter matters

Algebra is the largest and highest-weightage chapter in TN Class 10 Maths. Quadratic equations, polynomials and matrices carry many marks and build the foundation for higher-secondary mathematics.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

🔗

Class 9 — Polynomials and equations

Linear equations and polynomial basics.

Algebra — Class 10 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 10 Mathematics, Chapter 3 (the largest chapter). Equations, polynomials, quadratics and matrices.

1. About this chapter

This chapter covers simultaneous linear equations in three variables, GCD/LCM of polynomials, rational expressions, square root of polynomials, quadratic equations, and matrices.

2. Equations and polynomials

Simultaneous linear equations in three variables: solved by elimination/substitution to find x, y, z.
GCD and LCM of polynomials: related by f(x) · g(x) = GCD × LCM.
Rational expressions: ratios of polynomials; add, subtract, multiply and divide like fractions.
Square root of polynomials: found by factorisation or the long-division method.

3. Quadratic equations

Standard form: ax² + bx + c = 0 (a ≠ 0).
Solve by factorisation, completing the square, or the quadratic formula: x = [ −b ± √(b² − 4ac) ] / 2a.
Discriminant Δ = b² − 4ac decides the nature of roots:
- Δ > 0 → real and distinct; Δ = 0 → real and equal; Δ < 0 → no real roots.
Sum of roots = −b/a; Product of roots = c/a.

4. Matrices

A matrix is a rectangular array of numbers (order = rows × columns).
Operations: addition/subtraction (same order), scalar multiplication, matrix multiplication (columns of first = rows of second), and transpose Aᵀ.

5. Worked examples

Example 1. Find the nature of roots of x² − 4x + 4 = 0. Δ = (−4)² − 4(1)(4) = 16 − 16 = 0 → real and equal roots.

Example 2. Find the sum and product of the roots of 2x² − 5x + 3 = 0. Sum = −b/a = 5/2; Product = c/a = 3/2.

Example 3. Solve x² − 5x + 6 = 0 by factorisation. (x − 2)(x − 3) = 0 → x = 2 or 3.

6. Common mistakes

Mistake: Forgetting a ≠ 0 in a quadratic. Fix: If a = 0 the equation is linear, not quadratic.
Mistake: Wrong sign in the formula. Fix: x = [ −b ± √(b² − 4ac) ] / 2a — note the −b and 2a.
Mistake: Multiplying matrices of incompatible order. Fix: AB exists only if columns of A = rows of B.

7. Practice (book-back style)

Write the quadratic formula.
Find the discriminant of 3x² + 5x − 2 = 0 and the nature of its roots.
Find the sum and product of the roots of x² − 7x + 10 = 0.
Solve x² + 2x − 8 = 0 by factorisation.
State the condition for the product AB of two matrices to exist.

8. Answer key

x = [ −b ± √(b² − 4ac) ] / 2a.
Δ = 25 − 4(3)(−2) = 25 + 24 = 49 > 0 → real and distinct.
Sum = 7, Product = 10.
(x + 4)(x − 2) = 0 → x = −4 or 2.
The number of columns of A must equal the number of rows of B.

9. Quick revision

Chapter 3 · equations, polynomials, quadratics, matrices.
f(x)·g(x) = GCD × LCM (polynomials).
Quadratic: x = [−b ± √(b² − 4ac)]/2a; Δ = b² − 4ac.
Δ > 0 distinct, = 0 equal, < 0 no real roots.
Sum of roots = −b/a, product = c/a; AB needs cols(A) = rows(B).

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Quadratic formula

x = [ −b ± √(b² − 4ac) ] / 2a

For ax² + bx + c = 0, a ≠ 0.

Discriminant

Δ = b² − 4ac

>0 distinct, =0 equal, <0 no real roots.

Roots relations

sum = −b/a, product = c/a

From the coefficients.

Polynomial GCD–LCM

f(x) · g(x) = GCD × LCM

Like numbers.

Matrix product condition

cols(A) = rows(B)

For AB to exist.

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting a ≠ 0 in a quadratic

✓ If a = 0 the equation is linear, not quadratic.

WATCH OUT

✗ Wrong sign in the quadratic formula

✓ x = [ −b ± √(b² − 4ac) ] / 2a — note the −b and 2a.

WATCH OUT

✗ Multiplying matrices of incompatible order

✓ AB exists only if columns of A equal rows of B.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on quadratics, polynomials and matrices.

Questions

Short answer questions

Discriminant, roots, GCD/LCM and rational expressions.

Questions

Long answer problems

Three-variable equations, quadratics and matrix operations.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Recall

Write the quadratic formula.

▶ Show solution

x = [ −b ± √(b² − 4ac) ] / 2a.

Q2MEDIUM· Numerical

Find the discriminant of 3x² + 5x − 2 = 0 and the nature of its roots.

▶ Show solution

Δ = 25 − 4(3)(−2) = 49 > 0 → real and distinct roots.

Q3EASY· Numerical

Find the sum and product of the roots of x² − 7x + 10 = 0.

▶ Show solution

Sum = 7, Product = 10.

Q4MEDIUM· Numerical

Solve x² + 2x − 8 = 0 by factorisation.

▶ Show solution

(x + 4)(x − 2) = 0 → x = −4 or 2.

Q5EASY· Concept

State the condition for the product AB of two matrices to exist.

▶ Show solution

The number of columns of A must equal the number of rows of B.

Q6MEDIUM· Numerical

Find the nature of roots of x² − 4x + 4 = 0.

▶ Show solution

Δ = 16 − 16 = 0 → real and equal roots.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 3 of Samacheer Kalvi Class 10 Mathematics (largest chapter).
•f(x)·g(x) = GCD × LCM for polynomials.
•Quadratic: x = [−b ± √(b² − 4ac)]/2a; Δ = b² − 4ac.
•Δ > 0 distinct, = 0 equal, < 0 no real roots.
•Sum of roots = −b/a, product = c/a.
•Matrix product AB needs cols(A) = rows(B).

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 10-15 marks across MCQ, short answer and long problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-3	Discriminant, roots, matrices
Short Answer	2-3	2-3	Quadratics, GCD/LCM, rational expressions
Long Answer	5	1-2	Three-variable equations, matrices

Prep strategy

Master the quadratic formula and discriminant
Practise GCD/LCM and square root of polynomials
Drill matrix multiplication and transpose
Learn sum and product of roots

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Modelling

Quadratic equations model projectile motion and areas.

Computer graphics

Matrices transform and rotate images.

Engineering

Systems of equations solve circuit and structural problems.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write a ≠ 0 and identify a, b, c
Use the discriminant before solving where asked
Check matrix orders before multiplying
Show full working in long-answer problems

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Form a quadratic equation given the sum and product of its roots.
Prove (A B)ᵀ = Bᵀ Aᵀ for suitable matrices.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN SSLC Class 10 Public ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

The discriminant Δ = b² − 4ac reveals the nature of the roots without solving: positive gives two distinct real roots, zero gives equal roots, and negative gives no real roots.

No. In general AB ≠ BA, and sometimes only one of the products even exists depending on the orders of the matrices.

Practice more, ask doubts, find a tutor

Tamil Nadu (TNBSE) Class 10 mock tests

Tamil Nadu (TNBSE) Class 10 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Algebra usually read these next.

Relations and Functions

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 1, Relations and Functions. Ordered pairs and the Cartesian product, relations, functions and their types (one-one, onto, bijective), representation of functions, and composition of functions. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

Numbers and Sequences

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 2, Numbers and Sequences. Euclid's division lemma and algorithm, the fundamental theorem of arithmetic, modular arithmetic, arithmetic and geometric progressions with their sums, and special series. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

Geometry

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 4, Geometry. Similarity of triangles, the Basic Proportionality (Thales) theorem and the angle bisector theorem, the Pythagoras theorem and its converse, the concurrency theorems (Ceva and Menelaus), and tangents and secants of circles. Notes with theorems, worked examples, common mistakes, book-back practice and quick revision.

Coordinate Geometry

Samacheer Kalvi (TN State Board) Class 10 Maths Chapter 5, Coordinate Geometry. Area of a triangle and a quadrilateral, the condition for collinear points, slope and inclination, and the equation of a straight line in different forms. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

🧑‍🏫