Complex Numbers and Quadratic Equations

Study Notes

Complex Numbers

Definition: A complex number is expressed in the form ( a + bi ), where:
- ( a ) is the real part.
- ( b ) is the imaginary part.
- ( i ) is the imaginary unit, defined as ( i^2 = -1 ).
Types of Complex Numbers:
- Real Numbers: When ( b = 0 ) (e.g., ( 3 + 0i )).
- Imaginary Numbers: When ( a = 0 ) (e.g., ( 0 + 2i )).
- Purely Imaginary Numbers: When ( a = 0 ) and ( b \neq 0 ).
Modulus and Argument:
- Modulus: ( |z| = \sqrt{a^2 + b^2} )
- Argument: ( \theta = \tan^{-1}\left(\frac{b}{a}\right) )
Operations:
- Addition: ( (a + bi) + (c + di) = (a+c) + (b+d)i )
- Subtraction: ( (a + bi) - (c + di) = (a-c) + (b-d)i )
- Multiplication: ( (a + bi)(c + di) = (ac - bd) + (ad + bc)i )
- Division: ( \frac{a + bi}{c + di} = \frac{(a+bi)(c-di)}{c^2 + d^2} )

Quadratic Equations

Standard Form: A quadratic equation is of the form ( ax^2 + bx + c = 0 ), where ( a \neq 0 ).
Solutions: Obtained using:
- Factoring: When possible, factor the equation.
- Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
- Completing the Square: Convert to the form ( (x - p)^2 = q ).
Discriminant: ( D = b^2 - 4ac )
- If ( D > 0 ): Two distinct real roots.
- If ( D = 0 ): One real root (repeated).
- If ( D < 0 ): No real roots (complex roots).

Linear Inequalities

Definition: An inequality involving linear expressions. Example: ( ax + b < 0 ).
Types of Inequalities:
- Single Variable: Solve for ( x ) in expressions like ( 2x + 3 > 5 ).
- Multi-variable: Inequalities like ( 2x + 3y < 6 ).
Solution Methods:
- Graphical Method: Graph the boundary line (e.g., ( ax + b = 0 )) and shade the appropriate region.
- Algebraic Method: Isolate the variable and determine the range of values satisfying the inequality.
Properties:
- If you multiply or divide both sides by a negative number, reverse the inequality sign.
- Solutions can be expressed in interval notation (e.g., ( (a, b) ) or ( [a, b] )).

Complex Numbers

A complex number has the form ( a + bi ), where:
- ( a ) represents the real part.
- ( b ) represents the imaginary part.
- ( i ) is the imaginary unit, with ( i^2 = -1 ).
Types of complex numbers include:
- Real Numbers: Occurs when ( b = 0 ) (e.g., ( 3 + 0i )).
- Imaginary Numbers: Occurs when ( a = 0 ) (e.g., ( 0 + 2i )).
- Purely Imaginary Numbers: Occurs when ( a = 0 ) and ( b \neq 0 ).
Modulus and argument of a complex number:
- The modulus is given by ( |z| = \sqrt{a^2 + b^2} ).
- The argument ( \theta ) can be found using ( \theta = \tan^{-1}\left(\frac{b}{a}\right) ).
Operations on complex numbers:
- Addition: Combine real and imaginary parts, ( (a + bi) + (c + di) = (a+c) + (b+d)i ).
- Subtraction: Similar to addition but subtract real and imaginary parts, ( (a + bi) - (c + di) = (a-c) + (b-d)i ).
- Multiplication: Use the distributive property, ( (a + bi)(c + di) = (ac - bd) + (ad + bc)i ).
- Division: Use the conjugate to simplify, ( \frac{a + bi}{c + di} = \frac{(a+bi)(c-di)}{c^2 + d^2} ).

Quadratic Equations

Standard quadratic equation form is ( ax^2 + bx + c = 0 ), with ( a \neq 0 ).
Solutions can be determined by:
- Factoring: When feasible, factor the quadratic expression.
- Quadratic Formula: Given as ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Completing the Square: Rearranging to ( (x - p)^2 = q ).
Discriminant ( D ) is calculated as ( D = b^2 - 4ac ):
- ( D > 0 ): Indicates two distinct real roots.
- ( D = 0 ): Indicates one real root (it is repeated).
- ( D < 0 ): Indicates no real roots, resulting in complex roots.

Linear Inequalities

A linear inequality is expressed as a relationship involving linear expressions (e.g., ( ax + b < 0 )).
Types of inequalities include:
- Single Variable: Solving for ( x ) in expressions like ( 2x + 3 > 5 ).
- Multi-variable: Involves inequalities with multiple variables, such as ( 2x + 3y < 6 ).
Solution methods encompass:
- Graphical Method: Graph the boundary line determined by ( ax + b = 0 ) and shade the applicable region.
- Algebraic Method: Isolate the variable and find the range satisfying the inequality.
Important properties to remember:
- Multiplying or dividing both sides by a negative number alters the inequality's direction.
- Solutions can be represented in interval notation (e.g., ( (a, b) ) for open intervals or ( [a, b] ) for closed intervals).

Description

This quiz covers the fundamentals of complex numbers, including their definition, types, operations, and properties such as modulus and argument. It also delves into quadratic equations, focusing on their standard form and solutions. Test your understanding of these essential algebra concepts!