Numbers and algebraic identities are among the most scoring and formula-driven topics in Quantitative Aptitude. But many students find them confusing because numbers behave differently when written in expanded form, simplified form, factored form, or polynomial form.

If you understand these three ideas:
✔ How expressions expand
✔ How expressions factorize
✔ How algebraic identities shorten long calculations

Then you can solve ANY algebraic question effortlessly.

This Numbers – Basic Formulae guide covers all identities, concepts, shortcuts, cases, explanations, examples, and exam tips, making it the perfect one-stop resource for competitive exams.

Quick Overview: Algebraic Identities Formulas

Concept / Situation	Considered (Values / Terms)	Used (Operation / Purpose)	Formula (With Meaning of Symbols Inside Row)
Difference of Squares	Two terms a, b	Convert product to simplified difference	(a + b)(a – b) = a² – b² (a, b = any real numbers)
Square of Sum	Two terms a, b	Expand square without multiplication	(a + b)² = a² + b² + 2ab (a = first term, b = second term)
Square of Difference	Two terms a, b	Expand square when subtraction involved	(a – b)² = a² + b² – 2ab (a = first term, b = second term)
Square of (a + b + c)	Three terms a, b, c	Expand trinomial square	(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Sum of Cubes	Two terms a, b	Factorize cubic sum	a³ + b³ = (a + b)(a² – ab + b²)
Difference of Cubes	Two terms a, b	Factorize cubic difference	a³ – b³ = (a – b)(a² + ab + b²)
General 3-Term Cubic Identity	Three variables a, b, c	Expand or factor 3-variable cubic	a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)
Special Case Formula	When a + b + c = 0	Convert cubic expression instantly	a³ + b³ + c³ = 3abc

Formulas for Algebraic Identities

Understanding algebraic identities is the first step to solving number-based or expression-based questions accurately. Competitive exams test whether students recognize the right pattern, not whether they can multiply manually.

Below are each formula explained in the same deep, step-by-step way as your Problems on Trains format.

1. Difference of Squares Formula

(a + b)(a – b) = a² – b²

This identity helps simplify multiplication of two expressions quickly.
Instead of expanding manually, you jump directly to square values.

Why this formula works

(a + b)(a – b) expands as:

a(a – b) + b(a – b)
= a² – ab + ab – b²
= a² – b²

Here, +ab and –ab cancel, leaving only squares of a and b.

Common Mistakes Students Make

• Expanding terms unnecessarily
• Forgetting that middle terms remove each other
• Confusing it with (a + b)²

Key Tip

Whenever expressions appear as “sum × difference”, apply the identity instantly.

2. Square of Sum Formula

(a + b)² = a² + b² + 2ab

This gives the expansion for the square of a binomial.

Why this formula works

(a + b) × (a + b)
= a² + ab + ab + b²
= a² + b² + 2ab

When to Use

• Squaring 2 numbers quickly
• Expanding expressions
• Recognizing perfect square patterns

Common Errors

• Forgetting 2ab
• Writing (a² + b²) only

3. Square of Difference Formula

(a – b)² = a² + b² – 2ab

This is similar to the previous identity but the sign of 2ab becomes negative.

Why this formula works

(a – b)(a – b)
= a² – ab – ab + b²
= a² + b² – 2ab

Where used

• Simplifying expressions
• Quick mental math
• Recognizing patterns in algebra

4. Square of Three-Term Sum Formula

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Why do we add all pair products?

The expansion includes:

• Every square (a², b², c²)
• Every pair multiplied twice (2ab, 2bc, 2ca)

When to Use

• Expanding trinomial squares
• Factorization
• Shortcut simplifications

Common Mistakes

• Missing one pair product
• Writing only 1(ab) instead of 2(ab)

5. Sum of Cubes Formula

a³ + b³ = (a + b)(a² – ab + b²)

Why this formula works

This formula breaks a cube-sum into a product of a simple binomial and a quadratic term.

Where used

• Polynomial factorization
• Solving cubic equations
• Simplifying expressions

6. Difference of Cubes Formula

a³ – b³ = (a – b)(a² + ab + b²)

Why this formula works

It follows the same structure as the sum of cubes but changes sign pattern inside.

Common Errors

• Using wrong middle signs
• Mixing up sum and difference of cubes

7. Three Variable Cubic Formula

a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ac)

This identity is powerful in advanced algebra and competitive exams.

Why we subtract 3abc

To balance all mixed terms in the expansion.

When to Use

• Simplifying heavy expressions
• Factorizing 3-term polynomials
• Solving special condition problems

8. Special Case: When a + b + c = 0

a³ + b³ + c³ = 3abc

Why this works

If (a + b + c) = 0, then in the full cubic formula:

(a + b + c) × (something)
becomes
0 × (something) = 0

So the remaining term becomes 3abc.

Common Clues That a+b+c=0

• Numbers like 3, –1, –2
• Expressions designed to cancel
• Exam trick questions

Smart Tips and Practical Tricks for Solving Algebraic Identities

Mastering algebraic identities becomes simple when you understand how expressions expand, factorize, and transform. Most students make mistakes not because formulas are difficult, but because they apply them without understanding the structure of the expression.
This section breaks down the most important concepts into clear, actionable tips so you can solve identity-based questions faster and more accurately.

1. Identify the Pattern Before Applying Any Formula

Every identity has a unique structure. Recognizing that structure instantly tells you which formula to use. Most students expand blindly, which leads to unnecessary steps and mistakes.

Common patterns:

• (a + b)² → Square of a sum
• (a – b)² → Square of a difference
• a² – b² → Difference of squares
• a³ + b³ → Sum of cubes
• a³ – b³ → Difference of cubes
• a³ + b³ + c³ – 3abc → Special three-variable identity

Once you spot the pattern, the correct identity becomes obvious.

2. Avoid Full Expansion When Factorization Makes the Job Easier

When you see expressions like a³ – b³ or a² – b²:
Do not multiply them out step-by-step.
Use the identity directly to save time.

Examples:

• a³ – b³ → (a – b)(a² + ab + b²)
• a² – b² → (a + b)(a – b)

Shortcut thinking greatly reduces calculation time.

3. Always Check If the Expression Matches “a + b + c = 0” Condition

Many exam questions secretly use this pattern.

If a + b + c = 0, then the entire expression:

a³ + b³ + c³ – 3abc
becomes

a³ + b³ + c³ = 3abc

This eliminates long calculations and leads straight to the answer.

Signs of hidden pattern:

• Numbers that cancel each other
• Expressions designed to sum to zero
• Tricky cubic identities

Recognizing this instantly gives you full marks.

4. Group Terms Smartly Before Applying Identities

Many polynomial expressions look complicated only because terms are scattered.
Grouping like terms reveals the correct identity.

Example:
x² + 6x + 9 → (x + 3)²
a² – b² → (a + b)(a – b)

Proper grouping prevents misapplication of formulas.

5. Memorize Formulas by Logic, Not by Rote Learning

Every identity comes from multiplication rules.
If you understand how (a + b)(a + b) expands, you never need to ‘memorize’ (a + b)².

Logical understanding ensures you don’t confuse:

• (a + b)² and (a – b)²
• a³ + b³ and a³ – b³
• a² – b² and (a + b)²

Focus on patterns → not on rote.

6. Never Mix Identities from Different Categories

A very common exam mistake:

Using square formula for cubic expressions
Using cube formula for square expressions
Expanding wrongly grouped terms

Always check:
✔ Is it a square?
✔ Is it a cube?
✔ Is it a two-term or three-term expression?

Clarity before calculation prevents errors.

7. Practice Standard Exam Patterns Regularly

Identity questions appear in almost every aptitude exam:

• SSC (CGL, CHSL, CPO, GD)
• Railway RRB
• Banking (IBPS, SBI)
• State PSC
• CAT, MAT, Campus Placement Aptitude

Practicing these ensures you instantly recognize whether an expression is:

• Square formula
• Cube formula
• Special identity
• Factorization pattern
• Expansion case

The more patterns you know, the faster you solve.

FAQs About Algebraic Identities

Q1. Why are algebraic identities important in competitive exams?

They simplify expressions quickly and reduce long multiplications, saving time in aptitude tests.

Q2. How do I know which identity to apply?

Identify the pattern: squares, cubes, differences, or special cases. The structure of the expression tells you the formula.

Q3. Why is the identity a³ + b³ + c³ – 3abc so important?

It quickly simplifies complex cubic expressions, especially when a + b + c = 0.

Q4. What is the easiest identity to apply during exams?

(a + b)(a – b) = a² – b², because it simplifies multiplication instantly.

Q5. Why do students confuse (a + b)² and (a – b)²?

Both formulas look similar but differ in the sign of 2ab. Lack of pattern recognition causes mistakes.

Q6. How does checking a + b + c = 0 help in solving questions faster?

It turns a large expression into a simple one:
a³ + b³ + c³ = 3abc
This shortcut eliminates multi-step calculations.

Q7. Can algebraic identities be used for fast mental math?

Yes. Squares and cube formulas help in quick squaring, expansion, and factorization.

Q8. Why is factorization preferred over expansion?

Factorization reduces the number of steps and prevents large calculations, lowering chances of error.

Q9. What is the best way to master cube formulas?

Practice recognizing expressions in the form a³ ± b³. Pattern familiarity makes them automatic.

Q10. Why do students go wrong in identity questions?

They mix formulas, expand unnecessarily, or fail to identify the pattern correctly. Consistent practice removes confusion.