In short: Algebraic identities start with simple expansions like (a+b)^2 in Class 8 and grow into the binomial theorem, quadratic formula, sequences, matrices, and determinants by Class 12. This complete reference sheet covers every important identity and formula across five years of school maths, organised by class level. Use it for quick revision before exams, during homework, or any time you need to verify an expansion.
Why Algebraic Identities Follow You for Five Years
Algebra is the language of mathematics. The identity (a+b)^2 = a^2 + 2ab + b^2 that you learn in Class 8 does not disappear. It shows up in Class 9 polynomials, Class 10 quadratic equations, Class 11 binomial expansions, and Class 12 determinant problems. Each year adds new identities on top of the ones you already know.
The student who memorises (a+b)^2 in Class 8 and forgets it by Class 10 will struggle with completing the square in quadratic equations. The student who remembers it can derive the quadratic formula from scratch. That is the power of knowing your identities cold.
This sheet is organised by class level so you can see exactly which identities you should know at each stage.
Class 8 - Basic Algebraic Identities
These are the first algebraic identities you learn. They appear in every exam from Class 8 through competitive exams like JEE.
Square Identities
| Identity | Expansion |
|---|---|
| (a + b)^2 | a^2 + 2ab + b^2 |
| (a - b)^2 | a^2 - 2ab + b^2 |
| a^2 - b^2 | (a + b)(a - b) |
| (a + b + c)^2 | a^2 + b^2 + c^2 + 2ab + 2bc + 2ca |
Useful Rearrangements
| Formula | Derived From |
|---|---|
| a^2 + b^2 = (a + b)^2 - 2ab | Rearranging (a+b)^2 |
| a^2 + b^2 = (a - b)^2 + 2ab | Rearranging (a-b)^2 |
| (a + b)^2 - (a - b)^2 = 4ab | Subtracting the two identities |
| (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) | Adding the two identities |
Exam tip: The identity a^2 - b^2 = (a+b)(a-b) is used in factorisation problems across Classes 8, 9, and 10. It also appears in rationalisation of surds (Class 9) and in simplifying limits (Class 11).
Class 9 - Cube Identities and Polynomials
Class 9 extends square identities to cubes and introduces polynomial factorisation.
Cube Identities
| Identity | Expansion |
|---|---|
| (a + b)^3 | a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b) |
| (a - b)^3 | a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a - b) |
| a^3 + b^3 | (a + b)(a^2 - ab + b^2) |
| a^3 - b^3 | (a - b)(a^2 + ab + b^2) |
| a^3 + b^3 + c^3 - 3abc | (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) |
Special Case
| Condition | Result |
|---|---|
| If a + b + c = 0 | Then a^3 + b^3 + c^3 = 3abc |
Why this matters: The condition a + b + c = 0 leading to a^3 + b^3 + c^3 = 3abc is a favourite board exam question. It appears in both Class 9 and Class 10 exams almost every year.
Polynomial Identities (Class 9)
| Formula | Use |
|---|---|
| Remainder theorem | When polynomial p(x) is divided by (x - a), remainder = p(a) |
| Factor theorem | If p(a) = 0, then (x - a) is a factor of p(x) |
| Degree of product | deg(p x q) = deg(p) + deg(q) |
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Class 10 - Quadratic Equations and Arithmetic Progressions
Quadratic Equation Formulas
For the standard form ax^2 + bx + c = 0:
| Formula | Expression |
|---|---|
| Quadratic formula | x = (-b +/- sqrt(b^2 - 4ac)) / (2a) |
| Discriminant | D = b^2 - 4ac |
| Sum of roots | alpha + beta = -b/a |
| Product of roots | alpha x beta = c/a |
| Equation from roots | x^2 - (sum of roots)x + (product of roots) = 0 |
Nature of Roots
| Discriminant Value | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots |
| D < 0 | No real roots (complex roots) |
| D > 0 and is a perfect square | Rational roots |
Arithmetic Progression (AP) Formulas
| Formula | Expression |
|---|---|
| nth term | a(n) = a + (n - 1)d |
| Sum of n terms | S(n) = n/2 [2a + (n - 1)d] |
| Sum of n terms (alternate) | S(n) = n/2 [a + l], where l = last term |
| Common difference | d = a(n) - a(n-1) |
| Middle term (odd n) | a((n+1)/2) = Average of first and last terms |
Board exam favourite: "Find the sum of first 100 natural numbers" uses S = n(n+1)/2 = 100 x 101/2 = 5050. This is a special case of the AP sum formula where a = 1 and d = 1.
Class 11 - Binomial Theorem, Permutations, and Sequences
Permutations and Combinations
| Formula | Expression |
|---|---|
| Factorial | n! = n x (n-1) x (n-2) x ... x 1, and 0! = 1 |
| Permutations | nPr = n! / (n - r)! |
| Combinations | nCr = n! / (r!(n - r)!) |
| nCr = nC(n-r) | Symmetry property |
| nC0 + nC1 + ... + nCn | 2^n |
| nC0 = nCn | 1 |
| nCr + nC(r+1) | (n+1)C(r+1) - Pascal's rule |
Binomial Theorem
| Formula | Expression |
|---|---|
| Binomial expansion | (a + b)^n = Sum from r=0 to n of nCr a^(n-r) b^r |
| General term (r+1)th | T(r+1) = nCr a^(n-r) b^r |
| (1 + x)^n expansion | 1 + nC1 x + nC2 x^2 + ... + nCn x^n |
| Middle term (n even) | T(n/2 + 1) |
| Middle terms (n odd) | T((n+1)/2) and T((n+3)/2) |
Special Binomial Results
| Identity | Value |
|---|---|
| (1 + 1)^n = 2^n | Sum of all binomial coefficients |
| (1 - 1)^n = 0 | nC0 - nC1 + nC2 - ... = 0 |
| nC0 + nC2 + nC4 + ... | 2^(n-1) (sum of even-indexed terms) |
| nC1 + nC3 + nC5 + ... | 2^(n-1) (sum of odd-indexed terms) |
Sequences and Series
| Series | nth Term | Sum of n Terms |
|---|---|---|
| AP | a + (n-1)d | n/2 [2a + (n-1)d] |
| GP | a r^(n-1) | a(r^n - 1)/(r - 1) for r not equal to 1 |
| GP (infinite, | r | < 1) |
| Sum of first n naturals | - | n(n+1)/2 |
| Sum of squares of n naturals | - | n(n+1)(2n+1)/6 |
| Sum of cubes of n naturals | - | [n(n+1)/2]^2 |
Arithmetic Mean and Geometric Mean
| Formula | Expression |
|---|---|
| Arithmetic Mean of a, b | AM = (a + b)/2 |
| Geometric Mean of a, b | GM = sqrt(ab) |
| AM >= GM | Always true for positive numbers |
Complex Numbers (Class 11)
| Formula | Expression |
|---|---|
| i = sqrt(-1) | i^2 = -1, i^3 = -i, i^4 = 1 |
| Modulus of z = a + bi | |
| Conjugate of z = a + bi | z(bar) = a - bi |
| z x z(bar) | a^2 + b^2 = |
| Argument | arg(z) = tan^(-1)(b/a) |
Linear Inequalities (Class 11)
| Property | Rule |
|---|---|
| Adding same number | If a < b, then a + c < b + c |
| Multiplying by positive | If a < b and c > 0, then ac < bc |
| Multiplying by negative | If a < b and c < 0, then ac > bc (inequality reverses) |
Class 12 - Matrices, Determinants, and Advanced Algebra
Matrix Operations
| Operation | Formula |
|---|---|
| Scalar multiplication | k[a(ij)] = [ka(ij)] |
| Matrix addition | [a(ij)] + [b(ij)] = [a(ij) + b(ij)] |
| Transpose property | (AB)^T = B^T A^T |
| (A^T)^T | A |
| (A + B)^T | A^T + B^T |
| (kA)^T | k(A^T) |
Determinant Formulas
2x2 matrix:
| det(A) for [[a, b], [c, d]] | ad - bc |
|---|
3x3 matrix (expansion along first row):
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
for matrix [[a, b, c], [d, e, f], [g, h, i]]
Determinant Properties
| Property | Statement |
|---|---|
| Rows and columns swapped | det unchanged |
| Two rows/columns identical | det = 0 |
| Row/column multiplied by k | det multiplied by k |
| det(kA) for n x n matrix | k^n det(A) |
| det(AB) | det(A) x det(B) |
| det(A^T) | det(A) |
| det(A^(-1)) | 1/det(A) |
Inverse of a Matrix
| Formula | Expression |
|---|---|
| Condition for inverse | det(A) is not equal to 0 |
| A^(-1) (2x2) | (1/det(A)) x [[d, -b], [-c, a]] |
| A^(-1) (general) | (1/det(A)) x adj(A) |
| A x A^(-1) | I (identity matrix) |
| (AB)^(-1) | B^(-1) A^(-1) |
Cramer's Rule (System of Linear Equations)
For the system a1x + b1y = c1, a2x + b2y = c2:
| Variable | Formula |
|---|---|
| D (determinant of coefficients) | a1 b2 - a2 b1 |
| x | D1/D where D1 replaces a-column with c-column |
| y | D2/D where D2 replaces b-column with c-column |
Probability Formulas (Class 11-12)
| Formula | Expression |
|---|---|
| P(A) | Number of favourable outcomes / Total outcomes |
| P(A') (complement) | 1 - P(A) |
| P(A or B) | P(A) + P(B) - P(A and B) |
| P(A and B) for independent events | P(A) x P(B) |
| Conditional probability | P(A |
| Bayes' theorem | P(A |
| Total probability | P(B) = Sum of P(B |
| Mean of random variable | E(X) = Sum of x_i P(x_i) |
| Variance | Var(X) = E(X^2) - [E(X)]^2 |
| Binomial probability | P(X = r) = nCr p^r q^(n-r), where q = 1-p |
| Mean of binomial | np |
| Variance of binomial | npq |
Cross-Class Quick Reference - Most Used Identities
These are the identities that appear most frequently across all exams from Class 8 to 12. If you memorise nothing else, memorise these.
| # | Identity | First Taught |
|---|---|---|
| 1 | (a + b)^2 = a^2 + 2ab + b^2 | Class 8 |
| 2 | (a - b)^2 = a^2 - 2ab + b^2 | Class 8 |
| 3 | a^2 - b^2 = (a+b)(a-b) | Class 8 |
| 4 | (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 | Class 9 |
| 5 | a^3 + b^3 = (a+b)(a^2 - ab + b^2) | Class 9 |
| 6 | a^3 - b^3 = (a-b)(a^2 + ab + b^2) | Class 9 |
| 7 | Quadratic formula: x = (-b +/- sqrt(D)) / 2a | Class 10 |
| 8 | AP: a(n) = a + (n-1)d | Class 10 |
| 9 | GP sum: S = a(r^n - 1)/(r-1) | Class 11 |
| 10 | Binomial: (a+b)^n = Sum nCr a^(n-r) b^r | Class 11 |
How to Study Algebraic Identities Effectively
1. Understand Before Memorising
Every identity can be proved. (a+b)^2 = a^2 + 2ab + b^2 comes from multiplying (a+b) by itself. If you understand the multiplication, you will never forget the result. Rote memorisation without understanding breaks down under exam pressure.
2. Practise in Both Directions
Board exams test identities in two ways: expansion (given (a+b)^2, expand it) and factorisation (given a^2 + 2ab + b^2, recognise it as (a+b)^2). Students who only practise expansion struggle with factorisation. Do both.
3. Use Numerical Verification
After writing an identity, verify it with numbers. For (a+b)^2 = a^2 + 2ab + b^2, put a = 3, b = 2. Left side: (3+2)^2 = 25. Right side: 9 + 12 + 4 = 25. This verification habit catches errors before they reach your answer sheet.
4. Connect Across Classes
When you learn the binomial theorem in Class 11, verify that putting n = 2 gives you the Class 8 identity (a+b)^2. Putting n = 3 gives (a+b)^3 from Class 9. Seeing these connections makes the binomial theorem feel like a natural extension, not a new formula to memorise.
5. Write Them Out by Hand
Motor memory is powerful. Writing (a+b)^2 = a^2 + 2ab + b^2 by hand activates different neural pathways than just reading it. Before an exam, write out the 10 most-used identities from memory on a blank sheet. This 5-minute exercise can save you during the exam.
Common Mistakes Students Make
1. Sign errors in (a-b)^2. The expansion is a^2 - 2ab + b^2, not a^2 - 2ab - b^2. The last term is always positive because (-b)^2 = +b^2.
2. Confusing a^3 + b^3 and (a+b)^3. These are completely different. (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, while a^3 + b^3 = (a+b)(a^2 - ab + b^2). Mixing them up is one of the most common errors in Class 9 exams.
3. Forgetting the condition for AP/GP sum formulas. The GP sum formula S = a(r^n - 1)/(r-1) only works when r is not equal to 1. If r = 1, every term equals a and S = na.
4. Applying the quadratic formula with wrong sign. The formula is x = (-b +/- sqrt(D)) / 2a. Students often write (b +/- sqrt(D)) / 2a, missing the negative sign on b.
5. Determinant expansion errors. In 3x3 determinant expansion, the signs alternate: +, -, +. Many students forget the minus sign on the second term and get the wrong answer.
FAQ
How many algebraic identities do I need for Class 10 boards?
For Class 10 CBSE boards, you need approximately 15-20 identities: the basic square and cube identities from Class 8-9, the quadratic formula and its discriminant conditions, and AP formulas. These cover the direct algebra questions as well as the identities needed in coordinate geometry and statistics.
Do I need Class 8 identities for JEE?
Absolutely. JEE problems regularly use basic identities like a^2 - b^2 = (a+b)(a-b) inside complex problems. The difference is that JEE will not ask you to expand (a+b)^2 directly - it will embed that identity inside a limit, integral, or coordinate geometry problem.
What is the most important identity across all classes?
(a + b)^2 = a^2 + 2ab + b^2 and its sibling (a - b)^2 = a^2 - 2ab + b^2 are arguably the most used. They appear in completing the square for quadratic equations, in distance formulas, in deriving the quadratic formula, in proving AM-GM inequality, and in dozens of other contexts.
How do I know which identity to use in a problem?
Look at the structure. If you see a^2 + b^2 + 2ab, think (a+b)^2. If you see something^2 - something_else^2, think difference of squares. If you see x^3 + 8, recognise it as x^3 + 2^3 and apply the sum of cubes formula. Pattern recognition improves with practice.
Is the binomial theorem just a generalisation of (a+b)^2 and (a+b)^3?
Exactly. The binomial theorem gives you (a+b)^n for any positive integer n. When n = 2, you get the Class 8 identity. When n = 3, you get the Class 9 identity. When n = 10 or n = 50, you use the general formula with nCr coefficients.
Do matrices and determinants count as algebra?
Yes. Matrices and determinants are part of linear algebra, which is a branch of algebra dealing with systems of equations, transformations, and vector spaces. The determinant formulas and inverse matrix calculations in Class 12 are algebraic identities, just applied to arrays of numbers instead of single variables.
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