Complete Algebra Formula Guide: Essential Math Formulas for Students|All Algebra Formulas You Must Know – From Basics to Advanced

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VIDYA UNNATI ACCADEMY

Best Algebra Formulas Handbook – Quick Reference for Students.

Vidya Unnati Academy "Comprehensive Algebra Formulas," is a detailed collection of essential algebraic formulas, covering fundamental to advanced topics. It includes:

1. Basic Algebraic Identities – Expansions like , , and cube formulas.

2. Quadratic Equations – General form, roots formula, discriminant, and nature of roots.

3. Binomial Theorem – Expansion formula for any power .

4. Exponents & Logarithms – Power rules, logarithmic properties, and simplifications.

5. Polynomial Identities – Special polynomial factorization and expansion formulas.

6. Sequences & Series – Arithmetic and geometric progression formulas.

7. Matrices & Determinants – Basic determinant formulas, inverse, and multiplication rules.

8. Complex Numbers – Properties, modulus, conjugate, and De Moivre’s theorem.

This Post is useful for students preparing for exams like CBSE, ICSE, JEE, NEET, and competitive tests. It provides a quick reference for solving algebra.

Comprehensive Algebra Formulas

1. Basic Algebraic Identities

I. (a + b)² = a² + 2ab + b²

II. (a – b)² = a² - 2ab + b²

III. A² - b² = (a – b)(a + b)

IV. (x + a)(x + b) = x² + (a + b)x + ab

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

VI. (a – b – c)² = a² + b² + c² - 2(ab + bc + ca)

VII. A³ + b³ = (a + b)(a² - ab + b²)

VIII. A³ - b³ = (a – b)(a² + ab + b²)

IX. (a + b)³ = a³ + 3a²b + 3ab² + b³

X. (a – b)³ = a³ - 3a²b + 3ab² - b³

2. Quadratic Equations

I. General form: ax² + bx + c = 0

II. Roots formula: x = (-b ± √(b² - 4ac)) / 2a

III. Sum of roots: α + β = -b/a

IV. Product of roots: αβ = c/a

V. Quadratic Discriminant: Δ = b² - 4ac

VI. Nature of Roots: If Δ > 0 → Real and distinct roots, If Δ = 0 → Real and equal roots, If Δ < 0 → Imaginary roots

3. Binomial Theorem

I. (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n

4. Exponents and Logarithms

I. A^m * a^n = a^(m+n)

II. (a^m)ⁿ = a^(m*n)

III. A^0 = 1

IV. A^(-n) = 1/a^n

V. (ab)^n = a^n * b^n

VI. (a/b)^n = a^n / b^n

VII. Log(ab) = log a + log b

VIII. Log(a/b) = log a – log b

IX. Log(a^b) = b log a

X. Log(1) = 0

XI. Log(a) + log(b) = log(ab)

5. Polynomial Identities

I. (x – a)(x – b)(x – c)...(x – n) = 0

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

III. X³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy – yz – zx)

IV. (x + y + z)³ = x³ + y³ + z³ + 3(x + y)(y + z)(z + x)

V. (x – y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵

6. Sequences and Series

I. Sum of first n natural numbers: Sₙ = n(n + 1) / 2

II. Sum of squares of first n natural numbers: Sₙ = n(n + 1)(2n + 1) / 6

III. Sum of cubes of first n natural numbers: Sₙ = (n(n + 1)/2)²

IV. Arithmetic Progression (AP): aₙ = a + (n-1)d

V. Sum of AP: Sₙ = n/2 (2a + (n-1)d)

VI. Geometric Progression (GP): aₙ = ar^(n-1)

VII. Sum of GP: Sₙ = a(1 – rⁿ) / (1 – r), for r ≠ 1

VIII. Infinite GP sum: S = a / (1 – r), for |r| < 1

7. Matrices and Determinants

I. Determinant of 2×2 matrix: |A| = ad – bc for A = |a b| |c d|

II. Inverse of a 2×2 matrix: A⁻¹ = (1/|A|) * |d -b| | -c a|

III. Multiplication rule: (AB)⁻¹ = B⁻¹A⁻¹

IV. Identity Matrix: I * A = A * I = A

V. Transpose of a matrix: (Aᵀ)ᵀ = A

VI. Cofactor Expansion: |A| = Σ aᵢⱼ * Cᵢⱼ

8. Complex Numbers

I. I² = -1

II. Complex number: z = a + bi

III. Modulus: |z| = √(a² + b²)

IV. Conjugate: z̅ = a – bi

V. Multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i

VI. De Moivre’s Theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)

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