Complex Numbers and Quadratic Equations

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} )

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).

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] )).

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} ).

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.

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).