Algebra

Algebra: From Basics to Advanced Concepts

What is Algebra?
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It’s a unifying thread of almost all mathematics. Whether you’re solving a simple equation or constructing an advanced mathematical model, algebra is foundational.

Algebra helps in:

• Solving real-life problems.
• Creating mathematical models.
• Making predictions based on data.
• Building logic and analytical skills.

🟩 1. The Basics of Algebra

🔹 1.1 Variables and Constants

• Variables: Letters like x, y, z used to represent unknown values.
• Constants: Fixed values like 2, 5, -3, etc.

Example:

x+5=9x + 5 = 9x+5=9 → here, x is the variable, 5 and 9 are constants.

🔹 1.2 Algebraic Expressions

An expression is a combination of variables, constants, and operations (+, –, ×, ÷).

Examples:

• 3x+23x + 23x+2
• a2−4ab+b2a^2 – 4ab + b^2a2−4ab+b2

🔹 1.3 Terms, Coefficients, and Like Terms

• Term: A part of an expression separated by + or -.
• Coefficient: The number multiplying a variable.
• Like Terms: Same variables and exponents.

Example:

• In 7x+5y−3x7x + 5y – 3x7x+5y−3x, like terms are 7x7x7x and −3x-3x−3x.

🔹 1.4 Basic Operations

• Addition/Subtraction: Combine like terms.
• Multiplication: Apply distributive property.
• Division: Simplify fractions.

Example:

2(x+3)=2x+62(x + 3) = 2x + 62(x+3)=2x+6

🟨 2. Solving Equations

🔹 2.1 One-Step Equations

Solve x+5=12x + 5 = 12x+5=12

Subtract 5: x=7x = 7x=7

🔹 2.2 Two-Step Equations

Solve 2x−3=72x – 3 = 72x−3=7

Add 3: 2x=102x = 102x=10, Divide by 2: x=5x = 5x=5

🔹 2.3 Multi-Step Equations

Solve 3(x−2)+4=103(x – 2) + 4 = 103(x−2)+4=10

3x−6+4=103x – 6 + 4 = 103x−6+4=10 → 3x−2=103x – 2 = 103x−2=10 → 3x=123x = 123x=12 → x=4x = 4x=4

🔹 2.4 Equations with Variables on Both Sides

Solve 4x+2=2x+104x + 2 = 2x + 104x+2=2x+10

2x=82x = 82x=8 → x=4x = 4x=4

🟦 3. Inequalities

🔹 3.1 Solving Inequalities

x+3<7x + 3 < 7x+3<7 → x<4x < 4x<4

🔹 3.2 Graphing Inequalities

Use a number line. If inequality is strict (< or >), use open circle. If it includes equality (≤ or ≥), use closed circle.

🔹 3.3 Compound Inequalities

−2<x+1<3-2 < x + 1 < 3−2<x+1<3 → Subtract 1: −3<x<2-3 < x < 2−3<x<2

🟥 4. Polynomials

🔹 4.1 Introduction

Polynomials are expressions with many terms:

2×3+3×2−x+52x^3 + 3x^2 – x + 52×3+3×2−x+5

🔹 4.2 Degree of a Polynomial

• Highest power of the variable.
• x2+3x+5x^2 + 3x + 5×2+3x+5 → Degree = 2

🔹 4.3 Operations on Polynomials

• Addition/Subtraction: Combine like terms.
• Multiplication: Use distributive or FOIL method.
• Division: Long division or synthetic division.

🔹 4.4 Special Products

• (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2
• (a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2(a−b)2=a2−2ab+b2
• (a+b)(a−b)=a2−b2(a + b)(a – b) = a^2 – b^2(a+b)(a−b)=a2−b2

🟧 5. Factoring

🔹 5.1 Common Factoring

3x+6=3(x+2)3x + 6 = 3(x + 2)3x+6=3(x+2)

🔹 5.2 Difference of Squares

x2−9=(x+3)(x−3)x^2 – 9 = (x + 3)(x – 3)x2−9=(x+3)(x−3)

🔹 5.3 Trinomials

x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)x2+5x+6=(x+2)(x+3)

🔹 5.4 Factoring by Grouping

ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)

🟪 6. Rational Expressions

🔹 6.1 Simplifying

x2−9x+3=(x−3)(x+3)x+3=x−3\frac{x^2 – 9}{x + 3} = \frac{(x – 3)(x + 3)}{x + 3} = x – 3x+3×2−9​=x+3(x−3)(x+3)​=x−3

🔹 6.2 Operations

Add, subtract, multiply, divide like normal fractions with variables.

1x+1y=x+yxy\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}x1​+y1​=xyx+y​

🟫 7. Radicals and Exponents

🔹 7.1 Laws of Exponents

• am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n
• (am)n=amn(a^m)^n = a^{mn}(am)n=amn
• a0=1a^0 = 1a0=1

🔹 7.2 Simplifying Radicals

50=25⋅2=52\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}50​=25⋅2​=52​

🔹 7.3 Rationalizing the Denominator

12=22\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}2​1​=22​​

🟦 8. Quadratic Equations

🔹 8.1 Standard Form

ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

🔹 8.2 Solving Methods

• Factoring
• Completing the Square
• Quadratic Formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

🔹 8.3 Discriminant

D=b2−4acD = b^2 – 4acD=b2−4ac

• D>0D > 0D>0: 2 real roots
• D=0D = 0D=0: 1 real root
• D<0D < 0D<0: No real root

🟥 9. Functions and Graphs

🔹 9.1 Definition

A function assigns exactly one output for every input.

🔹 9.2 Function Notation

f(x)=2x+1f(x) = 2x + 1f(x)=2x+1

🔹 9.3 Graphing Linear Functions

Straight line: y=mx+by = mx + by=mx+b, where m is the slope, b is the y-intercept.

🔹 9.4 Quadratic Functions

Parabola shaped graph: y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c

🟨 10. Advanced Algebraic Topics

🔹 10.1 Logarithms

log⁡b(x)=y\log_b(x) = ylogb​(x)=y means by=xb^y = xby=x

Laws of logarithms:

• log⁡(ab)=log⁡a+log⁡b\log(ab) = \log a + \log blog(ab)=loga+logb
• log⁡(an)=nlog⁡a\log(a^n) = n \log alog(an)=nloga
• log⁡(ab)=log⁡a−log⁡b\log(\frac{a}{b}) = \log a – \log blog(ba​)=loga−logb

🔹 10.2 Exponential Functions

f(x)=a⋅bxf(x) = a \cdot b^xf(x)=a⋅bx

Used in growth, decay, population models, etc.

🔹 10.3 Systems of Equations

Solve multiple equations simultaneously using:

• Substitution
• Elimination
• Matrix method (Advanced)

🔹 10.4 Complex Numbers

i=−1i = \sqrt{-1}i=−1​, so i2=−1i^2 = -1i2=−1

• Standard form: a+bia + bia+bi

🟩 11. Real-Life Applications of Algebra

• Finance: Interest rates, EMIs, budgeting.
• Engineering: Calculations involving force, motion, current.
• Computer Science: Algorithms, logic gates, data structures.
• Physics: Equations of motion, optics, thermodynamics.
• Architecture: Geometry and structural calculations.

Algebra is more than just solving for “x.” It’s a powerful language that expresses relationships, solves complex problems, and enables discoveries in every field of science and engineering. Starting with basic operations and stretching all the way to functions and logarithms, mastering algebra opens the door to higher-level math and real-world problem solving.