Basic Algebra

Algebra is a branch of mathematics that uses symbols and letters (called variables) to represent numbers and express general relationships and rules.

It bridges arithmetic with advanced math by allowing you to:

• Represent unknown values with letters (like x, y)
• Form equations to describe real-world situations
• Simplify complex expressions using rules

Think of Algebra as arithmetic with placeholders. Instead of solving specific numbers, you solve for general cases.

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Key Concepts

Variables and Constants

• Variable: A symbol (usually x, y, etc.) representing an unknown or changing quantity.
• Constant: A fixed value (like 2, –7, ½, etc.).

Terms, Expressions, and Equations

• Term: A single quantity (e.g., 3x, –5, 7y²).
• Expression: A combination of terms using +, –, ×, ÷ (e.g., 3x + 5).
• Equation: An expression that equals another (e.g., 3x + 5 = 11).

Types of Algebraic Expressions

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Key Formulas & Identities

Fundamental Algebraic Identities

$$(x + y)^2 = x^2 + 2xy + y^2$$ $$(x - y)^2 = x^2 - 2xy + y^2$$ $$(x + y)(x - y) = x^2 - y^2$$ $$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

Cube Identities

$$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$$ $$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$$

Special Identities for Mental Calculations

$$(x + a)(x + b) = x^2 + (a + b)x + ab$$

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Algebraic Manipulations

Like Terms

Terms with the same variables and exponents can be combined.

Example:
3x + 5x = 8x

Distributive Law

$$a(b + c) = ab + ac$$

Factorization (Basic)

Factoring means expressing an expression as a product of its factors.

Common Factor:

$$2x + 4 = 2(x + 2)$$

Grouping:

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

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Conceptual Tips

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Visual Explanation

Think of an equation like balancing a scale:

• Whatever you do to one side, must be done to the other.
• Simplifying means grouping like shapes (terms) to make the scale readable.

Identity Visuals:

• $(x + y)^2$ is like a square with side length $x + y$
• $x^2 - y^2$ is the difference of two squares

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Examples

Example 1: Simplify an Expression

Q: Simplify:
$3x + 4 - 2x + 5$

A:
Group like terms:
= $(3x - 2x) + (4 + 5) = x + 9$

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Example 2: Use an Identity

Q: Evaluate $(a + b)^2$ when $a = 2, b = 3$

$$(a + b)^2 = a^2 + 2ab + b^2 = 4 + 12 + 9 = 25$$

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Example 3: Expand Using Identity

Q: Expand $(x - 2)^2$

$$x^2 - 4x + 4$$

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Example 4: Factor Using Identity

Q: Factor: $x^2 - 25$

$$x^2 - 5^2 = (x - 5)(x + 5)$$

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Example 5: Solve a Linear Equation

Q: Solve: $3x + 5 = 14$

$$3x = 14 - 5 = 9 \Rightarrow x = 3$$