Solving Quadratic Equations

Quadratic Equations

• 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.

Functions

• 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.

Inequalities

• 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)

Quadratic Equations

• 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.

Functions

• 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.

Inequalities

• 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)