Section 6.4: Equations

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6.4 Outline

  1. Terminology
    1. equation
      1. true
      2. false
      3. open
    2. satisfy
      1. root
      2. solution
      3. solve – this is the last of four main processes in algebra
    3. equivalent equations
    4. types
      1. linear equations
      2. quadratic equations
  2. Linear equations
    1. equation properties
      1. addition property
      2. subtraction property
      3. multiplication property
      4. division property
    2. goal of equation solving
    3. symmetric property of equality
    4. change a repeating decimal to fractional form
  3. Quadratic equations
    1. zero-product rule
    2. quadratic formula
    3. using technology to solve
    4. chaos and fractals

 

6.4 Essential Ideas

An equation is a statement of equality, and there are three types of equations:
open (has a variable), true, and false. To solve an equation is find the replacement(s) for the variable(s) that make the open equation a true equation. There are two types of equations solved in this section:

Linear equations: ax + b =0, (a not equal to 0)
Quadratic equations: ax2 + bx + c = 0 (a not equal to 0)

To solve a linear equation, you can use one or more of the following equation properties:

addition property: Adding the same number to both sides of an equation results in an equivalent equation.
subtraction property: Subtracting the same number from both sides of an equation results in a equivalent equation.
multiplication property: Multiplying both sides of a given equation by the same nonzero number results in an equivalent equation.
division property: Dividing both sides on a given equation by the same nonzero number results in an equivalent equation.

To solve a quadratic equation, use the following properties, in order:

zero-product rule: Obtain a zero on one side, and then factor the other side using the zero product rule.
quadratic formula: Write the equation in the form ax2 + bx + c = 0 (where a is not zero), and then use the quadratic formula.

Finally, algebraically simplify.