🎯 Algebra: Your Magic Toolbox for Solving Mysteries!
Imagine you’re a detective. You find clues (numbers and letters), follow patterns, and solve puzzles. That’s exactly what algebra is — being a math detective!
🌟 What is Algebra?
Think of algebra as a magic language where letters like x and y are treasure boxes. You don’t know what’s inside yet, but you can figure it out using clues!
Real Life Example:
- You have 5 apples. Your friend gives you some more. Now you have 8 apples.
- How many did your friend give? That’s algebra:
5 + x = 8, sox = 3!
📚 Table of Contents
- Linear Equations
- Quadratic Equations
- Inequalities
- Algebraic Identities
- Polynomials
- Arithmetic Progression
- Geometric Progression
- Sum of Progressions
1. Linear Equations
🎈 The Balancing Act
Imagine a seesaw at a playground. Both sides must be equal to balance. A linear equation is exactly that — keeping both sides equal!
What Makes It “Linear”?
The highest power of the variable is 1. No squares, no cubes — just simple!
Example: 2x + 5 = 11
How to Solve It
Step 1: Move the numbers to one side Step 2: Keep the variable (x) alone
2x + 5 = 11
2x = 11 - 5
2x = 6
x = 3 ✓
Real Life Story
You buy 2 identical toys and pay ₹5 extra for wrapping. Total = ₹11. How much is each toy? Answer: Each toy costs ₹3!
Types of Linear Equations
| Type | Example | Meaning |
|---|---|---|
| One Variable | 3x + 2 = 8 |
Find one unknown |
| Two Variables | x + y = 10 |
Find two unknowns |
graph TD A["Linear Equation"] --> B["Move constants to right"] B --> C["Divide by coefficient"] C --> D["Solution found!"]
2. Quadratic Equations
🎢 The Roller Coaster Curve
Quadratic equations create beautiful curves called parabolas — like the path of a ball you throw!
The Magic Formula
A quadratic equation looks like this:
ax² + bx + c = 0
Where a ≠ 0 (if a = 0, it becomes linear!)
Example Time!
x² - 5x + 6 = 0
Factoring Method:
- Find two numbers that multiply to 6 and add to -5
- Those are -2 and -3!
- (x - 2)(x - 3) = 0
- So x = 2 or x = 3 ✓
The Quadratic Formula (Your Secret Weapon!)
When factoring is hard, use this magic formula:
x = (-b ± √(b² - 4ac)) / 2a
Discriminant: The Fortune Teller
The part under the square root: b² - 4ac
| Value | What It Means |
|---|---|
| > 0 | Two different solutions |
| = 0 | One repeated solution |
| < 0 | No real solutions |
Real Life Story
A garden is shaped as a rectangle. Length is 3m more than width. Area is 28m². Find dimensions.
If width = x, then:
x(x + 3) = 28
x² + 3x - 28 = 0
x = 4 (width), x + 3 = 7 (length)
3. Inequalities
🚦 The Traffic Rules of Math
Equations say “equal.” Inequalities say “greater than” or “less than” — like speed limits!
The Four Symbols
| Symbol | Meaning | Example |
|---|---|---|
| > | Greater than | x > 5 |
| < | Less than | x < 10 |
| ≥ | Greater or equal | x ≥ 3 |
| ≤ | Less or equal | x ≤ 7 |
Solving Inequalities
Rule: Solve like equations, BUT flip the sign when multiplying or dividing by negative!
Example: -2x > 6
Divide by -2 (flip!):
x < -3
Number Line Visualization
x > 3: ----○========>
3
x ≥ 3: ----●========>
3
(○ = not included, ● = included)
Real Life Story
You need at least ₹500 for a trip. You have ₹200. How much more do you need?
200 + x ≥ 500
x ≥ 300
You need at least ₹300 more!
4. Algebraic Identities
🎩 Magic Shortcuts!
Identities are formulas that are always true. They’re like cheat codes for faster math!
The Famous Five
Identity 1: Perfect Square (Addition)
(a + b)² = a² + 2ab + b²
Example: (x + 3)² = x² + 6x + 9
Identity 2: Perfect Square (Subtraction)
(a - b)² = a² - 2ab + b²
Example: (x - 4)² = x² - 8x + 16
Identity 3: Difference of Squares
(a + b)(a - b) = a² - b²
Example: (x + 5)(x - 5) = x² - 25
Identity 4: Sum/Difference Product
(x + a)(x + b) = x² + (a+b)x + ab
Example: (x + 2)(x + 3) = x² + 5x + 6
Identity 5: Cube Formulas
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
Quick Calculation Trick!
Calculate 99² without a calculator:
99² = (100 - 1)²
= 100² - 2(100)(1) + 1²
= 10000 - 200 + 1
= 9801 ✓
5. Polynomials
🧱 Building Blocks of Algebra
A polynomial is like LEGO — different pieces (terms) connected together!
Anatomy of a Polynomial
3x³ + 2x² - 5x + 7
| Part | Name | Example |
|---|---|---|
| 3x³ | Term | Each piece |
| 3 | Coefficient | Number in front |
| 3 | Degree | Highest power |
| 7 | Constant | Number alone |
Types by Degree
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | 5 |
| 1 | Linear | 2x + 1 |
| 2 | Quadratic | x² + x |
| 3 | Cubic | x³ - 1 |
Types by Terms
| Terms | Name | Example |
|---|---|---|
| 1 | Monomial | 5x² |
| 2 | Binomial | x + 3 |
| 3 | Trinomial | x² + x + 1 |
Operations
Adding Polynomials — Combine like terms:
(2x² + 3x) + (x² - x) = 3x² + 2x
Multiplying Polynomials — Distribute each term:
(x + 2)(x + 3)
= x·x + x·3 + 2·x + 2·3
= x² + 5x + 6
6. Arithmetic Progression (AP)
🚶 Walking in Equal Steps
Imagine climbing stairs where each step is the same height. That’s an AP!
The Pattern
2, 5, 8, 11, 14, ...
Each number is 3 more than the previous. That 3 is called the common difference (d).
Key Formulas
First term: a Common difference: d nth term:
aₙ = a + (n - 1)d
Example: Find the 10th term
Given: First term = 3, common difference = 4
a₁₀ = 3 + (10 - 1) × 4
= 3 + 36
= 39 ✓
Finding d from the Sequence
Sequence: 7, 12, 17, 22, ...
d = 12 - 7 = 5
Real Life Story
You save ₹100 in week 1, ₹150 in week 2, ₹200 in week 3… How much in week 10?
a = 100, d = 50
a₁₀ = 100 + (9)(50) = ₹550
7. Geometric Progression (GP)
🚀 Growing by Multiplication
Instead of adding the same number, we multiply! Like bacteria doubling every hour.
The Pattern
2, 6, 18, 54, ...
Each number is 3 times the previous. That 3 is the common ratio ®.
Key Formulas
First term: a Common ratio: r nth term:
aₙ = a × r^(n-1)
Example: Find the 5th term
Given: First term = 2, common ratio = 3
a₅ = 2 × 3^(5-1)
= 2 × 3⁴
= 2 × 81
= 162 ✓
Finding r from the Sequence
Sequence: 5, 15, 45, 135, ...
r = 15 ÷ 5 = 3
AP vs GP: Quick Comparison
| Feature | AP | GP |
|---|---|---|
| Pattern | Add d | Multiply by r |
| Example | 2, 5, 8, 11 | 2, 6, 18, 54 |
| Growth | Linear | Exponential |
8. Sum of Progressions
💰 Counting Your Treasure
Now let’s add up all the terms! Like counting your total savings.
Sum of AP (Arithmetic Progression)
Formula for n terms:
Sₙ = n/2 × [2a + (n-1)d]
OR
Sₙ = n/2 × [first term + last term]
Example: Sum of first 10 terms where a = 2, d = 3
S₁₀ = 10/2 × [2(2) + (9)(3)]
= 5 × [4 + 27]
= 5 × 31
= 155 ✓
Famous Formula: Sum of 1 to n
1 + 2 + 3 + ... + n = n(n+1)/2
Sum of 1 to 100 = 100 × 101 ÷ 2 = 5050
Sum of GP (Geometric Progression)
When r ≠ 1:
Sₙ = a(rⁿ - 1)/(r - 1) [if r > 1]
Sₙ = a(1 - rⁿ)/(1 - r) [if r < 1]
Example: Sum of first 4 terms where a = 3, r = 2
S₄ = 3(2⁴ - 1)/(2 - 1)
= 3(16 - 1)/1
= 3 × 15
= 45 ✓
Infinite GP Sum (when |r| < 1)
S∞ = a/(1 - r)
The series goes on forever but has a finite sum!
Example: 1 + 1/2 + 1/4 + 1/8 + …
S∞ = 1/(1 - 0.5) = 1/0.5 = 2
🎯 Quick Summary
| Topic | Key Formula | Remember |
|---|---|---|
| Linear | ax + b = c | Balance like seesaw |
| Quadratic | ax² + bx + c = 0 | Makes parabola curve |
| Inequality | Flip sign with negative | Like speed limits |
| Identities | (a+b)² = a² + 2ab + b² | Magic shortcuts |
| Polynomials | Terms + Degree | LEGO blocks |
| AP | aₙ = a + (n-1)d | Equal steps |
| GP | aₙ = a × r^(n-1) | Multiply jumps |
| Sum AP | Sₙ = n/2 × (a + l) | Add savings |
| Sum GP | Sₙ = a(rⁿ - 1)/(r - 1) | Power growth |
🌈 Final Thoughts
You’ve just learned the entire toolkit of algebra! Remember:
- Linear equations = Balance the seesaw
- Quadratic equations = Throw and catch (parabolas)
- Inequalities = Follow the traffic rules
- Identities = Use your cheat codes
- Polynomials = Build with LEGO
- AP = Walk in equal steps
- GP = Jump and multiply
- Sums = Count your treasure
You’re now an Algebra Detective! Every problem is just a puzzle waiting to be solved. Go forth and conquer! 🚀