Solving Quadratic Equations

Study Notes

A quadratic equation is a polynomial equation of degree 2, with the general form: ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
Methods for solving quadratic equations:
- Factoring: if the equation can be written in the form (x - r)(x - s) = 0, then the solutions are x = r and x = s.
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, which works for all quadratic equations.
- Graphing: solving quadratic equations by graphing the related function on a coordinate plane.

A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
Key concepts:
- Domain: the set of all input values for which the function is defined.
- Range: the set of all possible output values of the function.
- Function notation: f(x) = ... , where f is the function name and x is the input variable.
- Composition of functions: (f ∘ g)(x) = f(g(x)), where f and g are functions.
Types of functions:
- Linear functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Exponential functions: f(x) = a^x, where a is the base and x is the exponent.

An inequality is a statement that compares two expressions using greater than, less than, greater than or equal to, or less than or equal to.
Key concepts:
- Graphing inequalities: solving inequalities by graphing the related function on a coordinate plane.
- Solving linear inequalities: using addition, subtraction, multiplication, and division to isolate the variable.
- Solving quadratic inequalities: using factoring, the quadratic formula, or graphing to solve.
Notation:
- > (greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)

A quadratic equation is a polynomial equation of degree 2, with the general form: ax^2 + bx + c = 0.
Factoring method: if the equation can be written in the form (x - r)(x - s) = 0, then the solutions are x = r and x = s.
Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, which works for all quadratic equations.
Graphing method: solving quadratic equations by graphing the related function on a coordinate plane.

A function is a relation between a set of inputs (domain) and a set of possible outputs (range).
Domain: the set of all input values for which the function is defined.
Range: the set of all possible output values of the function.
Function notation: f(x) = ..., where f is the function name and x is the input variable.
Composition of functions: (f ∘ g)(x) = f(g(x)), where f and g are functions.
Types of functions:
- Linear functions: f(x) = mx + b, where m is the slope and b is the y-intercept.
- Quadratic functions: f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Exponential functions: f(x) = a^x, where a is the base and x is the exponent.

An inequality is a statement that compares two expressions using greater than, less than, greater than or equal to, or less than or equal to.
Graphing inequalities: solving inequalities by graphing the related function on a coordinate plane.
Solving linear inequalities: using addition, subtraction, multiplication, and division to isolate the variable.
Solving quadratic inequalities: using factoring, the quadratic formula, or graphing to solve.
Notation:
- > (greater than)
- < (less than)
- ≥ (greater than or equal to)
- ≤ (less than or equal to)