Section 1-6:  Solving Quadratic Equations 
Any equation in the form
  ax2 + bx + c = 0 with a not equal 0 
is called a quadratic equation!
A value that satisfies this equation is called a root, zero or solution of the equation!!  We have three methods we can use to solve these equations:
  1)  factoring (easy)
   2)  completing the square
  3)  quadratic formula

Factoring review!
Solve each by factoring

 
Completing the square!!
Follow the explanation and sample problem to review completing the square
2x2 - 12x - 9 = 0

 
Quadratic Formula
As proved in class the quadratice formula is derived by completing the square.  Here is the formula:
 
If ax2 + bx + c = 0 then the roots of the equation are:
 
 
Look familiar?  It better!!
 

 
2x2 + 5 = 3x
                                              2x2 - 3x + 5 = 0 (putting in correct form)
a = 2, b = -3 and c = 5  Use the formula:
 
 

 
The Discriminant!
The quantity under the radical in the quadratic formula can tell us alot about the nature of the solutions.  Therefore, it is given a special name.  The discriminant is b2 - 4ac
If the discriminant is less than zero, then you will be taking the square root of a negative number yielding complex solutions.  If the discriminant equals zero, you have one real solution (namely -b/2a)  (Where have I heard that before?).  If the discriminant is greater than zero, then we have two different real solutions.  This is summarized in the following chart:
discriminant Types of solutions
b2 - 4ac < 0 Two complex conjugates
b2 -4ac = 0 One real (double root)
b2 - 4ac > 0 Two different real roots

 
Warning!  Warning!  Danger Ahead!!
Be extremely careful in solving problems like this:
x2 = x
It is very tempting to divide both sides by x to get:
x = 1
This is incorrect, because you have completely eliminated one of the solutions.  Never divide both sides of an equation by a variable if it cancels from both sides!!
Correct way to do the problem is as follows:
x2 = x
x2 - x = 0
x(x - 1) = 0
Solutions are x = 0 and x = 1
 

 
Honk!  Honk!  That about does it for this section!  Onward and upward!