Section 1.5 - Quadratic Equations

Section Objectives

  1. Solve quadratic equations by factoring.
  2. Solve quadratic equations by using square roots.
  3. Solve quadratic equations by using the quadratic formula.



A quadratic equation in the variable is an equation of the form , where . This is an example of a polynomial equation. In fact, it is a 2nd degree polynomial equation.



There are three common approaches to solving quadratic equations:

  1. Factor and set each factor to zero.
  2. Use square roots to "undo" the square (after completing the square).
  3. Use the quadratic formula.



Using Factoring to Solve Quadratic Equations

If several numbers (or expressions) are multiplied and their product is zero, then one of the factors must be zero. We normally state this property by saying:

If , then or .



This property allows us to solve quadratic (or higher-order) equations by factoring.



Examples



















Using Square Roots to Solve Quadratic Equations

A quadratic equation that has the form can be solved by taking square roots of both sides of the equation. Remember that a perfect square usually has two square roots, one positive and one negative.


If , then or .



Examples










Completing the Square

We can always use square roots to solve a quadratic equation as long as the square "is complete." To complete the square means to rewrite a standard form quadratic expression in terms of a complete square:



With practice, completing the square is usually pretty straight-forward. This sheet walks you through the process step by step. Refer to the sheet if you are having trouble.




Examples










Quadratic Formula

By completing the square, we can derive a formula that is capable giving the solutions to any quadratic equation. This formula is called the quadratic formula:


If , then .



The quantity under the radical, , is called the discriminant. Its sign tells us about the solutions:

  1. If , then there is only one real number solution.
  2. If , then there are two different real number solutions.
  3. If , then there are two different complex number solutions.


Examples