Equations

Algebraic Equations

Linear Equations

  • How to solve: Use techniques to rearrange the equation into the form x = c, where c is a constant.
  • Understand that the solution of the linear equation is the x-value that makes the equation true.
  • Recognize that linear equations will always have one solution, as long as there’s at least one variable with a non-zero coefficient.

Quadratic Equations

  • Definition: Quadratic equations are given in the form ax² + bx + c = 0. Here, a ≠ 0.
  • Key terms:
    • Roots: The solutions of the quadratic equation; the values that make the equation true.
    • Discriminant: The part underneath the radical in the quadratic formula (b² - 4ac). It determines the nature of the roots of the quadratic equation.
  • How to solve: Use factorising, completing the square, or using the quadratic formula.

Systems of Equations

  • Definition: Multiple equations with multiple variables that are solved simultaneously.
  • Key concept: A “solution” to a system of equations is a set of variable values that satisfies all of the equations in the system.
  • How to solve: Use substitution, elimination, or matrix methods.

Cubic Equations

  • Definition: Algebraic equations with maximum degree three; form of equation is ax³ + bx² + cx + d = 0.
  • Recognize that the solutions or roots of the cubic equations are the x-values that make the equation true.
  • Know that in the general cubic equation, if a ≠ 0, then it has at least one real root.

Differential Equations

  • Understand that a differential equation expresses a relationship between a function and its derivatives.
  • Be familiar with the basic method of solving first-order differential equations by separation of variables.