Class 9 Maths Chapter 4 Algebraic Identities

1. Revisiting Algebraic Identities

Algebraic identities are formulas that help us expand and factor expressions quickly. These are powerful tools for simplifying algebra.

\( (a + b)^2 = a^2 + 2ab + b^2 \)

\( (a - b)^2 = a^2 - 2ab + b^2 \)

\( (a + b)(a - b) = a^2 - b^2 \)

Identity	Expanded Form
Square of a sum	\( a^2 + 2ab + b^2 \)
Square of a difference	\( a^2 - 2ab + b^2 \)
Difference of squares	\( a^2 - b^2 \)

2. Visualising Identities Using Geometrical Models

Geometric models help us see why algebraic identities work. For example, \( (a+b)^2 \) can be seen as a square of side \( a+b \).

\( (a + b)^2 \) model

Area parts: \( a^2, b^2, 2ab \)

\( a^2 - b^2 \) model

Difference: Remove \( b^2 \) from \( a^2 \).

3. Factorisation Using Identities

Identities can be used to factor expressions fast. Convert the expression into a known identity and reverse the expansion.

Example: Factor \( x^2 + 5x + 6 \).

We use the identity \( a^2 + 2ab + b^2 = (a + b)^2 \) with \( a = x \) and \( b = 3 \), but first check if it fits.

The expression becomes \( x^2 + 2 \cdot x \cdot 3 + 3^2 = (x + 3)^2 \) only if the constant is 9, not 6. So instead factor directly:

\( x^2 + 5x + 6 = (x + 2)(x + 3) \).

Factor Finder

Try factoring: \( x^2 + 7x + 10 \)

4. More Identities and Their Applications

After the basic identities, there are more useful formulas that simplify algebra quickly.

Identity	Use
\( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \)	Expand cubes
\( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \)	Expand differences
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)	Factor cubic subtraction

Application: Use identities to rewrite sums, simplify expressions, and solve equations faster.

5. Visualising Factorisation of Quadratic Expressions

Algebra tiles show how a quadratic factors into two binomials. The area model explains the same idea without tiles.

Algebra Tiles

Boxes represent \( x^2, x, \) and constants.

x

2

x^2

Area Method

Show \( x^2 + 5x + 6 \) as a rectangle with sides \( x + 2 \) and \( x + 3 \).

6. Finding New Identities

New identities appear when you look for patterns. For example:

\( (a + b)^2 - (a - b)^2 = 4ab \)

This gives a new identity by comparing two expansions and cancelling terms.

Student Tip: Use decomposition and step-by-step reasoning to derive formulas from simple examples.

7. Simplifying Rational Expressions

Always factor numerator and denominator first, then cancel common factors.

\( \frac{x^2 - 9}{x^2 - 5x + 6} = \frac{(x - 3)(x + 3)}{(x - 2)(x - 3)} = \frac{x + 3}{x - 2} \)

Key idea: Simplification works only when the cancelled factor is not zero.

8. Learning Outcomes

Visualise algebraic identities using geometric models.
Determine the factors of algebraic expressions using identities.
Interpret factors of quadratic expressions through geometric models.
Find simplified versions of rational expressions.
Use computational thinking strategies such as decomposition and step-by-step procedures.

Concept Check Quiz

1. Which identity helps expand \( (a - b)^2 \)?

A) \( a^2 + b^2 \)

B) \( a^2 - 2ab + b^2 \)

C) \( a^2 + 2ab - b^2 \)

2. The expression \( x^2 - 4 \) factors as:

A) \( (x - 2)(x + 2) \)

B) \( (x - 4)(x + 1) \)

C) \( (x + 2)^2 \)

3. Simplify \( \frac{x^2 - 1}{x^2 - x} \).

A) \( \frac{x + 1}{x} \)

B) \( \frac{x - 1}{x} \)

C) \( \frac{x - 1}{x + 1} \)

4. The model for \( (a + b)^2 \) includes:

A) \( a^2, b^2, 2ab \)

B) \( a^2, b^2, ab \)

C) \( a^2, 2b^2, ab \)