Algebra Formulas for CBSE Class 8 to 12 — Identities, AP, GP, Logarithms
Algebra formulas include algebraic identities, the quadratic formula, arithmetic and geometric progressions, logarithm rules, binomial theorem, and permutations and combinations. They are covered in NCERT Class 8 through Class 12 and are essential for CBSE board exams, JEE Main, and JEE Advanced.
- Algebraic identities: squares, cubes, and polynomial identities
- Quadratic formula, discriminant, and nature of roots
- Arithmetic Progression (AP) — general term, sum, nth term
- Geometric Progression (GP) — finite and infinite sum
- Logarithm properties (log rules)
- Binomial theorem and Pascal’s triangle
- Permutations and combinations
What are Algebra Formulas?
Algebra formulas are equations that express the relationship between algebraic quantities using variables, constants, and operations. They allow us to expand, factorise, and simplify expressions without performing numerical calculations from scratch. Algebraic identities are true for all values of the variable. They are introduced from Class 8 (NCERT Chapter 9) and extended through Class 12.
Basic Algebraic Identities
These standard identities must be memorised. They are tested in CBSE Class 8, 9, and 10, and are used in Class 11 and 12 to simplify and factorise expressions.
| Identity Name | Formula | NCERT Class |
|---|---|---|
| Square of Sum | \( (a+b)^2 = a^2 + 2ab + b^2 \) | Class 8, Ch 9 |
| Square of Difference | \( (a-b)^2 = a^2 – 2ab + b^2 \) | Class 8, Ch 9 |
| Difference of Squares | \( a^2 – b^2 = (a+b)(a-b) \) | Class 8, Ch 9 |
| Square of Trinomial | \( (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \) | Class 9, Ch 2 |
| Cube of Sum | \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \) | Class 9, Ch 2 |
| Cube of Difference | \( (a-b)^3 = a^3 – 3a^2b + 3ab^2 – b^3 \) | Class 9, Ch 2 |
| Sum of Cubes | \( a^3 + b^3 = (a+b)(a^2 – ab + b^2) \) | Class 9, Ch 2 |
| Difference of Cubes | \( a^3 – b^3 = (a-b)(a^2 + ab + b^2) \) | Class 9, Ch 2 |
| Sum of Cubes (three vars) | \( a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \) | Class 9, Ch 2 |
Quadratic Formula and Nature of Roots
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). The roots are found using the quadratic formula.
| Concept | Formula |
|---|---|
| Quadratic Formula | \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) |
| Discriminant | \( D = b^2 – 4ac \) |
| Two distinct real roots | \( D > 0 \) |
| Two equal real roots | \( D = 0 \) |
| No real roots (complex) | \( D < 0 \) |
| Sum of roots | \( \alpha + \beta = -\frac{b}{a} \) |
| Product of roots | \( \alpha \beta = \frac{c}{a} \) |
Arithmetic Progression (AP) Formulas
An AP is a sequence where each term differs from the previous by a fixed value called the common difference (d).
| Concept | Formula |
|---|---|
| General (nth) term | \( a_n = a + (n-1)d \) |
| Sum of n terms | \( S_n = \frac{n}{2}(2a + (n-1)d) \) |
| Sum using first and last term | \( S_n = \frac{n}{2}(a + l) \) |
| Arithmetic Mean | \( A = \frac{a + b}{2} \) |
Geometric Progression (GP) Formulas
A GP is a sequence where each term is multiplied by a fixed ratio r.
| Concept | Formula |
|---|---|
| General (nth) term | \( a_n = ar^{n-1} \) |
| Sum of n terms (r ≠ 1) | \( S_n = \frac{a(r^n – 1)}{r – 1} \) |
| Sum of infinite GP (|r| < 1) | \( S_\infty = \frac{a}{1 – r} \) |
| Geometric Mean | \( G = \sqrt{ab} \) |
Logarithm Formulas
| Property | Formula |
|---|---|
| Product Rule | \( \log_a(mn) = \log_a m + \log_a n \) |
| Quotient Rule | \( \log_a\left(\frac{m}{n}\right) = \log_a m – \log_a n \) |
| Power Rule | \( \log_a(m^n) = n\log_a m \) |
| Change of Base | \( \log_a b = \frac{\log_c b}{\log_c a} \) |
| Identity | \( \log_a a = 1 \) |
| Zero property | \( \log_a 1 = 0 \) |
Permutations and Combinations
| Concept | Formula | NCERT |
|---|---|---|
| Permutation | \( ^nP_r = \frac{n!}{(n-r)!} \) | Class 11, Ch 7 |
| Combination | \( ^nC_r = \frac{n!}{r!(n-r)!} \) | Class 11, Ch 7 |
| Complement rule | \( ^nC_r = \,^nC_{n-r} \) | Class 11, Ch 7 |
| Binomial theorem | \( (a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k \) | Class 11, Ch 8 |
All Algebra Formula Articles
For official NCERT Maths textbooks from Class 8 to 12, visit ncert.nic.in. For CBSE exam pattern and sample papers, visit cbse.gov.in.