Class 10 Algebra Formulas – Complete List of Algebraic Identities & Key Formulas
Algebra forms a major portion of the CBSE Class 10 Mathematics syllabus. A strong command over algebraic identities and formulas helps students solve quadratic equations, polynomials, arithmetic progression problems, and pair of linear equations efficiently.
This article provides a complete and structured list of all important Class 10 Algebra formulas along with explanations and examples.
1. Algebraic Identities (Most Important)
Algebraic identities are formulas that are true for all values of variables. They are heavily used in factorization and expansion problems.
Basic Identities
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- (x + a)(x + b) = x² + (a + b)x + ab
Advanced Identities
- (a + b)³ = a³ + b³ + 3ab(a + b)
- (a − b)³ = a³ − b³ − 3ab(a − b)
- a³ + b³ = (a + b)(a² − ab + b²)
- a³ − b³ = (a − b)(a² + ab + b²)
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Example: Expand (2x + 3)²
= (2x)² + 2(2x)(3) + 3²
= 4x² + 12x + 9
2. Quadratic Equations Formulas
A quadratic equation is of the form:
ax² + bx + c = 0
Quadratic Formula
x = [-b ± √(b² − 4ac)] / 2a
Discriminant (D)
D = b² − 4ac
- If D > 0 → Two distinct real roots
- If D = 0 → Equal roots
- If D < 0 → No real roots
Example: Solve x² − 5x + 6 = 0
a = 1, b = -5, c = 6
D = (-5)² − 4(1)(6) = 25 − 24 = 1
x = [5 ± 1]/2
Roots = 3 and 2
3. Pair of Linear Equations in Two Variables
General form:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Consistency Conditions
- a₁/a₂ ≠ b₁/b₂ → Unique solution
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution
- a₁/a₂ = b₁/b₂ = c₁/c₂ → Infinitely many solutions
4. Polynomials Formulas
For quadratic polynomial ax² + bx + c:
Sum of zeros = -b/a
Product of zeros = c/a
Example: For 2x² − 7x + 3
Sum = 7/2
Product = 3/2
5. Arithmetic Progression (AP) Formulas
AP is a sequence where difference between consecutive terms is constant.
- nth term (aₙ) = a + (n − 1)d
- Sum of n terms (Sₙ) = n/2 [2a + (n − 1)d]
- Sₙ = n/2 (a + l)
Example: Find 10th term of AP: 3, 7, 11…
a = 3, d = 4
a₁₀ = 3 + 9×4 = 39
6. Important Factorization Formulas
- a² − b² = (a − b)(a + b)
- x² + (a + b)x + ab = (x + a)(x + b)
- a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
7. Linear Inequalities Basics
If sign changes when multiplying/dividing by negative number.
Example: If -2x > 4, then x < -2.
8. Key Exam-Oriented Shortcuts
✔ Always calculate discriminant first in quadratic equations
✔ In AP problems, identify a and d correctly
✔ Practice factorization using splitting middle term
Common Mistakes Students Make
- Forgetting minus sign in quadratic formula
- Incorrect expansion of (a − b)²
- Confusing sum and product of zeros
- Arithmetic errors in AP formula substitution
Final Revision Strategy
Revise algebra formulas daily before board exams. Practice 5 problems from each chapter: Quadratic Equations, Polynomials, AP, and Linear Equations. Formula clarity reduces calculation time and boosts accuracy.