Algebra Formulas: Complete List for Class 8, 9, 10, 11 and 12

Algebra formulas are standard equations that use variables, constants, and operations to express mathematical relationships. Students use algebra formulas across identities, quadratic equations, AP, GP, matrices, and vectors from Class 8 to 12.

Algebra is the branch of mathematics that replaces numbers with letters. Letters like x, y, a, and b represent unknown quantities. Formulas in algebra describe how these quantities relate to each other, making it possible to solve problems across every chapter from Class 8 to Class 12.

Knowing your algebra formulas saves time in exams. A student who has memorised (a + b)² can expand any square binomial in seconds. A student who knows the quadratic formula can solve any Class 10 polynomial without factoring. Extramarks covers the full list here from basic identities to matrices and JEE-level formulas.

Key Takeaways

What You Will Find Details
Classes Covered Class 8, 9, 10, 11, 12
Key Topics Identities, Quadratic Formula, AP, GP, Exponents, Logarithms, Matrices, Vectors
Quadratic Formula x = (−b ± √D) / 2a, where D = b² − 4ac
JEE Relevance Binomial Theorem, AP/GP, Matrices, Quadratic Equations
Total Formula Groups 10 chapterwise sections with solved examples

All Important Algebra Formulas: Quick Reference

These are the core algebraic identities every student from Class 8 onwards must know. Each formula below appears in CBSE board exams across multiple classes.

Extramarks algebra formulas chart featuring key identities and expansions in a clean orange and beige layout with numbered sections.

 

These identities form the foundation of all algebraic expressions and formulas across the CBSE syllabus. Explore the full list of Maths Formulas and build a foundation before moving to class-specific topics.

Algebraic Properties

Algebraic properties explain how variables behave during operations. These ideas support simplification, factorisation, and equation solving.

Commutative Property

  • Addition: a + b = b + a
  • Multiplication: ab = ba

Associative Property

  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (ab)c = a(bc)

Distributive Property

  • a(b + c) = ab + ac
  • a(b − c) = ab − ac

Identity Element

  • Addition: a + 0 = a
  • Multiplication: a × 1 = a

Inverse Element

  • Addition: a + (−a) = 0
  • Multiplication: a × 1/a = 1, where a ≠ 0

Algebra Formulas for Class 8

Algebra formulas for class 8 introduce students to algebraic expressions, identities, and variable-based expansion. These formulas build the base for Class 9 and 10.

The algebraic expressions and identities Class 8 formulas are:

  • (a + b)² = a² + 2ab + b²
  • (a − b)² = a² − 2ab + b²
  • (a + b)(a − b) = a² − b²
  • (x + a)(x + b) = x² + x(a + b) + ab
  • (a + b)³ = a³ + 3a²b + 3ab² + b³
  • (a − b)³ = a³ − 3a²b + 3ab² − b³
  • a³ + b³ = (a + b)(a² − ab + b²)
  • a³ − b³ = (a − b)(a² + ab + b²)
  • (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

The (a + b)² formula and the (a + b)(a − b) formula appear in almost every Class 8 and Class 9 exam. Practise them with numbers first, substitute a = 3 and b = 2 before using them with variables.

Algebra Formulas for Class 9

Class 9 extends the algebraic expressions formula list to higher powers and factorisation methods. It also introduces formulas that connect directly to exponent rules.

Strengthen your exam preparation with CBSE Important Questions Class 9 Maths to test how well you can apply these formulas under exam conditions.

  • a³ − b³ = (a − b)(a² + ab + b²)
  • a³ + b³ = (a + b)(a² − ab + b²)
  • (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
  • (a − b)⁴ = a⁴ − 4a³b + 6a²b² − 4ab³ + b⁴
  • (x³ + y³ + z³ − 3xyz) = (x + y + z)(x² + y² + z² − xy − yz − zx)
  • If (x + y + z) = 0, then x³ + y³ + z³ = 3xyz

Exponent Formulas

Laws of exponents are important from Class 9 onwards and remain useful in algebra, indices, and logarithms. 

  • aᵐ × aⁿ = aᵐ⁺ⁿ
  • aᵐ / aⁿ = aᵐ⁻ⁿ
  • a⁰ = 1
  • a⁻¹ = 1/a
  • aᵐ × bᵐ = (ab)ᵐ
  • (aᵐ)ⁿ = aᵐⁿ

Logarithm Formulas

  • log_a(xy) = log_a x + log_a y
  • log_a(x/y) = log_a x − log_a y
  • log_a(xᵐ) = m log_a x
  • log_a a = 1
  • log_a 1 = 0

Cross-check your working with NCERT Solutions Class 9 Maths for step-by-step solutions to every exercise.

Algebra Formulas for Class 10

Algebra formulas for class 10 are among the most tested formulas in board exams. The most important topics are the quadratic formula, discriminant, and AP.

Quadratic Formula and Discriminant

x = (−b ± √D) / 2a

Where D = b² − 4ac is called the discriminant.

Discriminant Value Nature of Roots
D > 0 Real and distinct
D = 0 Real and equal (coincident)
D < 0 Non-real (no real roots)

Sum and Product of Roots

For a quadratic ax² + bx + c = 0 with roots p and q:

  • Sum of roots: p + q = −b/a
  • Product of roots: pq = c/a

For a cubic polynomial ax³ + bx² + cx + d with roots p, q, r:

  • p + q + r = −b/a
  • pq + qr + pr = c/a
  • pqr = −d/a

The discriminant conditions are frequently tested in MCQ and VSA format. Memorise the three cases positive, zero, and negative as a table. Work through CBSE Important Questions Class 10 Maths to practise discriminant and root-based problems before your board exam.

AP and GP Formulas: Class 10 and 11

AP and GP formulas are important for Class 10, Class 11, and JEE. Students should revise the nth term and sum formulas together.

Arithmetic Progression (AP)

For a sequence: a, a+d, a+2d, …

  • Common difference: d = a₂ − a₁
  • nth term: aₙ = a + (n − 1)d
  • nth term from last: aₙ = l − (n − 1)d
  • Sum of first n terms: Sₙ = n/2 [2a + (n − 1)d]
  • Sum using first and last term: Sₙ = n/2 (a + l)

Geometric Progression (GP)

For a sequence: a, ar, ar², …

  • Common ratio: r = ar^(n−1) / ar^(n−2)
  • nth term: aₙ = ar^(n−1)
  • nth term from last: aₙ = l / r^(n−1)
  • Sum of first n terms (r < 1): Sₙ = a(1 − rⁿ) / (1 − r)
  • Sum of first n terms (r > 1): Sₙ = a(rⁿ − 1) / (r − 1)

Algebra Formulas for Class 11

Class 11 algebra formulas cover four major topics: inequalities, permutation and combination, binomial theorem, and GP extension. These also build the base for JEE preparation.

Inequality Formulas

  • If x > y and z > 0: xz > yz
  • If x > y and z < 0: xz < yz
  • If x > y: x + z > y + z and x − z > y − z

Permutation and Combination Formulas

  • P(n, r) = n! / (n − r)!
  • C(n, r) = n! / [(n − r)! × r!]
  • n! = n × (n − 1) × (n − 2) × … × 1
  • 0! = 1

Binomial Theorem

(a + b)ⁿ = aⁿ + (ⁿC₁)aⁿ⁻¹b + (ⁿC₂)aⁿ⁻²b² + … + (ⁿCₙ₋₁)abⁿ⁻¹ + bⁿ

The general term is: Tᵣ₊₁ = ⁿCᵣ × aⁿ⁻ʳ × bʳ

Factorials and Binomial Theorem are best studied together. The binomial theorem is essentially a generalised version of (a + b)² and (a + b)³ that students already know from Class 8.

Algebra Formulas for Class 12

Class 12 algebra moves into matrices, determinants, and vectors. These topics carry significant weightage in boards and JEE Main.

Matrix Formulas

  • Addition: If A and B are m × n matrices, C = A + B where cᵢⱼ = aᵢⱼ + bᵢⱼ
  • Multiplication: If A is m × n and B is n × p, then C = AB where cᵢⱼ = Σ aᵢₖbₖⱼ
  • Scalar multiplication: (kA)ᵢⱼ = k × aᵢⱼ
  • Transpose: (Aᵀ)ᵢⱼ = aⱼᵢ
  • Adjoint: adj(A) = Cᵀ (transpose of cofactor matrix)
  • Inverse: A⁻¹ = adj(A) / det(A), provided det(A) ≠ 0

Determinant Formulas

For a 2 × 2 matrix A = [[a, b], [c, d]]: det(A) = ad − bc

For a 3 × 3 matrix A = [[a, b, c], [d, e, f], [g, h, i]]: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Vector Formulas

  • Magnitude: |v| = √(vₓ² + vy² + vz²)
  • Unit vector: û = v / |v|
  • Dot product: a · b = aₓbₓ + ayby + azbz
  • Cross product: a × b = (aybz − azby)i − (axbz − azbx)j + (axby − aybx)k

Algebra Formulas for JEE

Algebra formulas for JEE come from several chapters together. These are the most important ones to revise first.

Topic Key Formula JEE Chapter
Quadratic Equations x = (−b ± √D) / 2a; D = b² − 4ac Quadratic Equations
Binomial Theorem (a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ Binomial Theorem
AP Sₙ = n/2 [2a + (n − 1)d] Sequences and Series
GP Sₙ = a(rⁿ − 1)/(r − 1) Sequences and Series
Matrices A⁻¹ = adj(A)/det(A) Matrices and Determinants
Permutation P(n, r) = n!/(n − r)! Permutations and Combinations
Combination C(n, r) = n!/[(n − r)! r!] Permutations and Combinations
Vectors a · b = a

JEE Main typically tests the quadratic discriminant, sum/product of roots, and binomial general term in every paper. Focus on those three first.

Solved Examples Using Algebra Formulas

Example 1: Find the value of (5x + 4)²

Solution: Using (a + b)² = a² + 2ab + b², where a = 5x and b = 4: (5x + 4)² = (5x)² + 2(5x)(4) + 4² = 25x² + 40x + 16

Example 2: Find the value of (9x − 5y)²

Solution: Using (a − b)² = a² − 2ab + b², where a = 9x and b = 5y: (9x − 5y)² = 81x² − 90xy + 25y²

Example 3: Calculate 205 × 195

Solution: Using (a + b)(a − b) = a² − b², write 205 × 195 = (200 + 5)(200 − 5): = 200² − 5² = 40000 − 25 = 39975

Example 4: For the equation 2x² − 7x + 3 = 0, find the nature of roots and the sum and product of roots.

Solution: D = b² − 4ac = (−7)² − 4(2)(3) = 49 − 24 = 25 D > 0, so roots are real and distinct. Sum of roots = −b/a = 7/2 Product of roots = c/a = 3/2

How to Remember Algebra Formulas

Students searching how to remember algebra formulas usually need a practical method, not only theory. These steps help with faster recall. 

  1. Derive each identity once on your own.
  2. Group formulas chapter-wise.
  3. Practise with number substitution first.
  4. Revise formula groups daily for one week.
  5. Solve 2 to 3 questions after each revision block.

FAQs (Frequently Asked Questions)

The basic algebra formulas every student must know are the core algebraic identities: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b², and the sum and difference of cubes. These cover most questions in Class 8, 9, and 10 board exams and form the base for Class 11 and 12 topics.

The quadratic formula is x = (−b ± √D) / 2a, where D = b² − 4ac is the discriminant. It gives the roots of any quadratic equation ax² + bx + c = 0. When D > 0, roots are real and distinct. When D = 0, roots are real and equal. When D < 0, there are no real roots. This formula applies to every quadratic equation regardless of whether it factors easily.

An algebraic expression is a combination of variables, constants, and operations, for example, 3x² + 2x − 5. An algebraic formula is an equation that shows a specific, verified relationship between expressions, for example, (a + b)² = a² + 2ab + b². Expressions describe quantities; formulas describe how those quantities relate.

AP formulas in Class 11: nth term aₙ = a + (n − 1)d; sum Sₙ = n/2[2a + (n − 1)d]. GP formulas: nth term aₙ = ar^(n−1); sum Sₙ = a(rⁿ − 1)/(r − 1) for r > 1. The key difference is that AP uses a common difference while GP uses a common ratio. Both topics appear in JEE Mains under Sequences and Series.

The highest-weightage algebra formulas for JEE are the quadratic discriminant and root conditions, the binomial theorem general term (Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ), AP and GP sum formulas, matrix determinant and inverse, and permutation-combination formulas. JEE Main tests at least 4 to 5 questions from these topics every year.

The laws of exponents in algebra are: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ᵐ = aᵐbᵐ; a⁰ = 1; a⁻ⁿ = 1/aⁿ. These rules apply to all real bases (with a ≠ 0 where required) and form the basis for logarithm rules in Class 9 and 11.