Algebra is the backbone of mathematics. Whether you are preparing for SSC, Banking, Railways, NDA, CDS, Teaching Exams, State PSC, or school exams, algebra plays a crucial role in scoring high marks.
In this detailed guide, you will find:
- ✅ All Basic Algebra Formulas
- ✅ Square & Cube Identities
- ✅ Laws of Exponents
- ✅ Linear & Quadratic Equations
- ✅ Factorization Formulas
- ✅ Simplified Explanations
- ✅ Practice Examples
- ✅ Official Educational References
📘 What is Algebra?
Algebra is a branch of mathematics that uses symbols (like x, y, z) to represent numbers and solve equations.
Instead of writing:
5 + some number = 12
We write:
5 + x = 12
This makes complex problems easier to solve.
📊 PART 1: Basic Algebra Formulas
1️⃣ Laws of Exponents (Power Rules)
If a and b are numbers:
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- (ab)ⁿ = aⁿbⁿ
- a⁰ = 1
- a⁻ⁿ = 1 / aⁿ
Example:
2³ × 2² = 2⁵ = 32
📐 PART 2: Algebraic Identities (Very Important)
These are extremely important for competitive exams.
1️⃣ (a + b)² Formula
[
(a + b)^2 = a^2 + 2ab + b^2
]
Example:
(5 + 3)² = 25 + 30 + 9 = 64
2️⃣ (a − b)² Formula
[
(a - b)^2 = a^2 - 2ab + b^2
]
3️⃣ (a + b)(a − b)
[
(a + b)(a - b) = a^2 - b^2
]
4️⃣ (a + b)³ Formula
[
(a + b)^3 = a^3 + b^3 + 3ab(a + b)
]
5️⃣ (a − b)³ Formula
[
(a - b)^3 = a^3 - b^3 - 3ab(a - b)
]
6️⃣ a³ + b³ Formula
[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
]
7️⃣ a³ − b³ Formula
[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
]
📈 PART 3: Linear Equations
General Form:
[
ax + b = 0
]
Solution:
[
x = -b/a
]
Example:
3x + 6 = 0
x = -6/3 = -2
📊 PART 4: Quadratic Equation Formula
Standard Form:
[
ax^2 + bx + c = 0
]
Quadratic Formula:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
The value inside the square root is called Discriminant (D)
[
D = b^2 - 4ac
]
If:
- D > 0 → Two real roots
- D = 0 → Equal roots
- D < 0 → No real roots
📏 PART 5: Factorization Formulas
Common Factor Method
Example:
6x + 12 = 6(x + 2)
Middle Term Splitting
Example:
x² + 5x + 6
= x² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
📉 PART 6: Algebraic Fractions
If:
[
\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}
]
📊 PART 7: Important Formulas for Competitive Exams
- Percentage → x% = x/100
- Profit = SP − CP
- Simple Interest = (P × R × T) / 100
- Average = Sum / Total Numbers
🎯 Tips to Master Algebra Quickly
✔ Practice daily
✔ Memorize identities
✔ Solve previous year papers
✔ Revise exponent rules
✔ Use shortcut tricks
📚 Official Educational References
For authentic syllabus and mathematical guidelines, refer to:
📘 National Council of Educational Research and Training (NCERT)
Official Website:
https://ncert.nic.in
📘 Central Board of Secondary Education (CBSE)
Official Website:
https://cbse.gov.in
📘 National Testing Agency (NTA)
Official Website:
https://nta.ac.in
📝 Practice Questions
- Expand (7 + 2)²
- Solve: 2x + 5 = 11
- Factorize: x² − 9
- Solve: x² − 5x + 6 = 0
📌 Common Mistakes Students Make
❌ Forgetting 2ab term
❌ Wrong sign in (a − b)²
❌ Calculation errors in quadratic formula
❌ Not simplifying fully
🏆 Why Algebra is Important?
- Required for almost all competitive exams
- Helps in higher mathematics
- Builds logical thinking
- Useful in real life calculations
📌 Quick Revision Chart
|
Formula |
Expression |
|
(a + b)² |
a² + 2ab + b² |
|
(a − b)² |
a² − 2ab + b² |
|
a² − b² |
(a + b)(a − b) |
|
a³ + b³ |
(a + b)(a² − ab + b²) |
|
Quadratic |
(-b ± √D) / 2a |
📍 Final Conclusion
Algebra becomes easy when formulas are understood—not memorized blindly. With consistent practice and proper concept clarity, you can score full marks in algebra sections of competitive exams.
Refer official academic resources for structured learning:
- NCERT → https://ncert.nic.in
- CBSE → https://cbse.gov.in
- NTA → https://nta.ac.in
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