What is the quadratic formula?

For ax^2 + bx + c = 0, the roots are x = [-b ± √(b^2 - 4ac)] / 2a. The part b^2 - 4ac is the discriminant.

How does the discriminant decide the nature of roots?

If b^2 - 4ac > 0 the roots are real and distinct; if = 0 they are real and equal; if < 0 there are no real roots (the roots are imaginary).

When can two matrices be multiplied?

Two matrices A and B can be multiplied only when the number of columns of A equals the number of rows of B.

Tamil Nadu Class 10 Maths Chapter 3 - Algebra

Q: What is a polynomial?

A polynomial is an algebraic expression with variables and coefficients in which every power of the variable is a non-negative integer, e.g. 3x^2 + 2x - 5.

Q: When can two matrices be multiplied?

Two matrices A and B can be multiplied only when the number of columns of A equals the number of rows of B.

Exercise Solutions Available

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Overview

Algebra is one of the highest-scoring Class 10 chapters because the methods are fixed — once you know the standard steps, every sum follows the same pattern. This chapter covers GCD/LCM of polynomials, algebraic identities, rational expressions, quadratic equations (three methods), and matrices.

1. GCD and LCM of Polynomials

GCD (HCF): the highest-degree expression that divides two or more polynomials exactly.
LCM: the lowest-degree expression that is a multiple of each polynomial.
Key relation: f(x) × g(x) = GCD × LCM. If you know any three, you can find the fourth.

Method: factorise each polynomial fully → GCD = product of common factors (lowest powers); LCM = product of all factors (highest powers).

2. Algebraic Identities

Identity	Expansion
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
(a + b)(a − b)	a² − b²
(a + b)³	a³ + 3a²b + 3ab² + b³
(a + b + c)²	a² + b² + c² + 2ab + 2bc + 2ca

Identities let you expand and, read in reverse, factorise quickly.

3. Quadratic Equations — three methods

A quadratic equation has the form ax² + bx + c = 0 (a ≠ 0).

Method 1 — Factorisation

Split the middle term so the two parts multiply to a·c and add to b.

Method 2 — Quadratic Formula

x = [ −b ± √(b² − 4ac) ] / 2a

Method 3 — Completing the square

Rewrite as (x + p)² = q and take square roots.

Nature of roots (discriminant Δ = b² − 4ac)

Δ > 0  → real and distinct roots
Δ = 0  → real and equal roots
Δ < 0  → no real roots (imaginary)

4. Matrices

Order = rows × columns. Add/subtract only when orders are equal.
Multiplication: A(m×n) × B(n×p) is possible (inner numbers match) and gives an m×p matrix, using the row-by-column rule.

Worked Examples

Example 1 — Factorisation

Solve x² − 5x + 6 = 0.

Split −5x as −2x − 3x (since (−2)(−3)=6, −2−3=−5):
x² − 2x − 3x + 6 = 0
x(x − 2) − 3(x − 2) = 0
(x − 2)(x − 3) = 0  →  x = 2 or x = 3

Example 2 — Quadratic formula

Solve 2x² + 3x − 5 = 0.

a = 2, b = 3, c = −5
Δ = 3² − 4(2)(−5) = 9 + 40 = 49  (>0 → real, distinct)
x = [−3 ± √49]/4 = [−3 ± 7]/4
x = 1   or   x = −5/2

Example 3 — Nature of roots

Find the nature of roots of x² − 6x + 9 = 0.

Δ = (−6)² − 4(1)(9) = 36 − 36 = 0  →  real and equal roots (x = 3, 3)

Example 4 — Matrix multiplication

A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]. Find AB.

AB = [[1·5+2·7, 1·6+2·8], [3·5+4·7, 3·6+4·8]]
   = [[19, 22], [43, 50]]

Example 5 — Word problem (quadratic)

The product of two consecutive positive integers is 56. Find them.

Let the numbers be x and x+1: x(x+1) = 56
x² + x − 56 = 0 → (x + 8)(x − 7) = 0 → x = 7 (positive)
Numbers: 7 and 8

Common Mistakes

Forgetting the middle term: (a + b)² is a² + 2ab + b², not a² + b².
Sign slips while splitting the middle term in factorisation.
Quoting only one root of a quadratic — always give both.
Multiplying matrices whose inner dimensions do not match.

Exam Tips & Revision Checklist

Memorise all identities and the quadratic formula.
For "nature of roots", compute the discriminant first.
Show every step — method marks are awarded even if the final answer slips.
Practised: GCD/LCM, factorisation, formula method, discriminant, matrix multiply.

Frequently Asked Questions

A polynomial is an algebraic expression with variables and coefficients in which every power of the variable is a non-negative integer, e.g. 3x^2 + 2x - 5.