Identities and Expansion

Understand important algebraic identities and how to expand expressions using them. Learn formulas like (a+b)², (a-b)², and (a+b)(a-b) with step-by-step examples.

1. Introduction

Algebraic identities are formulas that are always true for all values of the variables. These identities help us expand expressions quickly without multiplying each term manually. Knowing these identities saves time and makes calculations easier.

In this topic, we learn the most commonly used algebraic identities and how to apply them to expand expressions.

2. What Are Algebraic Identities?

An algebraic identity is a formula that holds true for every value of the variable(s). Identities are used to simplify expressions and expand brackets.

Examples of simple identities:

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

3. Standard Algebraic Identities

The following algebraic identities are most widely used in middle school algebra:

3.1. Identity 1: Square of a Sum

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

This identity helps expand expressions where a sum is squared.

3.1.1. Worked Examples

  • \((x + 3)^2 = x^2 + 2 \cdot x \cdot 3 + 9 = x^2 + 6x + 9\)
  • \((2a + 5)^2 = 4a^2 + 20a + 25\)

3.2. Identity 2: Square of a Difference

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

This identity is similar to the square of a sum but with a negative middle term.

3.2.1. Worked Examples

  • \((x - 4)^2 = x^2 - 8x + 16\)
  • \((3a - 2)^2 = 9a^2 - 12a + 4\)

3.3. Identity 3: Product of Sum and Difference

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

This identity gives the difference of squares.

3.3.1. Worked Examples

  • \((x + 5)(x - 5) = x^2 - 25\)
  • \((3y + 2)(3y - 2) = 9y^2 - 4\)

3.4. Identity 4: General Expansion

The distributive property helps expand products:

\((a + b)(c + d) = ac + ad + bc + bd\)

3.4.1. Worked Examples

  • \((x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6\)
  • \((2a + 1)(a - 4) = 2a^2 - 8a + a - 4 = 2a^2 - 7a - 4\)

4. Using Identities for Expansion

Identities make expansion simpler and faster. Instead of multiplying each term manually, you can directly apply the formula.

4.1. When to Use an Identity

Use an identity when the expression matches a known pattern.

  • Use \((a + b)^2\) when you see a square of a sum.
  • Use \((a - b)^2\) when you see a square of a difference.
  • Use \((a + b)(a - b)\) when you have a sum multiplied by a difference.

4.2. When to Use Direct Expansion

If the expression does not match any identity exactly, expand using distributive property.

  • Example: \((a + b + c)(a - b)\) does not match any simple identity.
  • So we expand each term: \(a(a - b) + b(a - b) + c(a - b)\).

5. Mixed Examples

Let’s use identities to expand the following:

5.1. Example Set

1. Expand \((3x + 2)^2\)

Using identity: \(a^2 + 2ab + b^2\)

  • \(a = 3x\)
  • \(b = 2\)

Result: \(9x^2 + 12x + 4\)

2. Expand \((y - 7)^2\)

Using: \(a^2 - 2ab + b^2\)

Result: \(y^2 - 14y + 49\)

3. Expand \((m + 4)(m - 4)\)

Using: \(a^2 - b^2\)

Result: \(m^2 - 16\)

6. Quick Practice

Expand using identities:

  1. \((x + 6)^2\)
  2. \((2a - 5)^2\)
  3. \((3p + 1)(3p - 1)\)
  4. \((a + b)(a + 2b)\) — use general expansion

7. Summary

  • Algebraic identities are formulas that hold true for all values of variables.
  • Common identities include \((a + b)^2\), \((a - b)^2\), and \((a + b)(a - b)\).
  • Use identities for quick expansion.
  • If identity patterns do not match, use the distributive property.