Algebra Fundamentals

Algebra is the branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations. It provides methods for solving problems that involve unknown values and understanding relationships between variables.

Basic Algebraic Concepts

Variables and Constants

A variable is a symbol (usually a letter) that represents an unknown number or value. A constant is a fixed value that doesn't change.

Example: In the expression 3x + 5:

  • x is the variable.
  • 3 and 5 are constants.

Algebraic Expressions

An algebraic expression combines variables, constants, and operations such as addition, subtraction, multiplication, and division.

Example: 4y^2 - 7y + 9 is an algebraic expression with terms involving powers of y.

Solving Linear Equations

A linear equation is an equation of the first degree, meaning the highest power of the variable is 1. The general form is:

ax + b = c

where a, b, and c are constants, and x is the variable.

Example 1: Simple Linear Equation

Solve for x: 3x + 5 = 20

  1. Subtract 5 from both sides: 3x = 20 - 5
  2. Simplify the right side: 3x = 15
  3. Divide both sides by 3: x = \frac{15}{3} = 5

Answer: x = 5

Example 2: Variables on Both Sides

Solve for x: 5x + 3 = 2x + 15

  1. Subtract 2x from both sides: 3x + 3 = 15
  2. Subtract 3 from both sides: 3x = 12
  3. Divide both sides by 3: x = 4

Answer: x = 4

Quadratic Equations

A quadratic equation is a polynomial equation of degree 2. Its general form is:

ax^2 + bx + c = 0

where a \ne 0.

Solving Using Quadratic Formula

The roots of a quadratic equation can be found using the formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Example: Solve 2x^2 - 4x - 6 = 0.

  1. Identify coefficients: a=2, b=-4, c=-6
  2. Calculate discriminant: D = b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 + 48 = 64
  3. Find square root of discriminant: \sqrt{64} = 8
  4. Calculate roots:
    • x = \frac{-(-4) + 8}{2 \times 2} = \frac{4 + 8}{4} = 3
    • x = \frac{-(-4) - 8}{2 \times 2} = \frac{4 - 8}{4} = -1

Answer: x = 3 or x = -1

Polynomials

A polynomial is an algebraic expression made up of variables and coefficients involving only addition, subtraction, multiplication, and non-negative integer exponents of variables.

General form of a polynomial in one variable x:

a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

where a_n, a_{n-1}, \ldots, a_0 are constants and n is a non-negative integer.

Example

4x^3 - 3x^2 + 2x - 5 is a polynomial of degree 3.

Functions

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.

It can be written as f(x), which means the function f depends on variable x.

Example

f(x) = 2x + 3

For each value of x, you get a corresponding output f(x).

Practice Problems

Problem 1: Solve for x: 4x - 7 = 9

  1. Add 7 to both sides: 4x = 16
  2. Divide both sides by 4: x = 4

Answer: x = 4

Problem 2: Solve for x: 3x + 5 = 2x + 11

  1. Subtract 2x from both sides: x + 5 = 11
  2. Subtract 5 from both sides: x = 6

Answer: x = 6

Problem 3: Solve the quadratic equation x^2 - 5x + 6 = 0

  1. Factor the equation: (x - 2)(x - 3) = 0
  2. Set each factor equal to zero:
    • x - 2 = 0 \Rightarrow x = 2
    • x - 3 = 0 \Rightarrow x = 3

Answer: x = 2 or x = 3