1.08  Quadratic Equations

College Algebra: One Step at a Time.  Pages 122-124:    #7

                                                                                 Pages 125 -129:   #11, 12

NEW PROBLEM:  #7

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

When solving a quadratic equation, there are several methods to consider.  Some problems can be solved best by one method, and other problems are best solved by different methods. 

Of course, the easiest way to solve any quadratic equation is the method of factoring--that is, of course, IF the equation can be factored.  Not all quadratic equations can be factored. 

The most reliable method is the quadratic formula: 

If ,   

then .

The quadratic formula ALWAYS works!

However, while the quadratic formula can be used to solve any problem, the completing the square method is sometimes easier.  Specifically, in order to solve by the completing the square method, the coefficient of  must be 1 in order for “half and square” to work.  Secondly, the completing the square method works very nicely when the x coefficient is an even number (so taking “half and square” doesn’t end up with horrible fractions to deal with!). 

                                                                   Completing the Square

p. 124.  #7.  Solve for x by completing the square:  .

Solution:                       

Set up for completing the square by adding 15 to each side:

Next, take “half and square”:

Square root both sides:

Simplify :                  

Add  +5  to each side:                        

NOTE:       If you do NOT have POLYSMLT on your TI 83+ or TI 84 calculator in the APPS menu of your calculator, it can be easily installed for you if you know someone who has it on their calculator! 

If you have a TI 86 or TI 85, the program you need is built into the calculator at [2^nd]   [POLY]. 

                                             Quadratic Formula

p. 129.  # 11.        

This is a quadratic equation, so the first step is to set it equal to zero, preferably with a positive   term.  This can be done by moving everything to the right side, by adding and  to each side.

This quadratic equation cannot be solved by factoring, so in this case, the quadratic formula is the best choice of methods.

Remember that the solution to the quadratic equation:   

                                                           is given by the formula:   

In the equation:  

Factor out the common factor of 2 in the numerator, and reduce the fraction.

p. 129.  # 12.   

The first step is to multiply out the parentheses, and notice that it is a quadratic equation.  When you see this, you might as well go ahead and set it equal to zero by adding +5 to each side of the equation.

This quadratic equation cannot be solved by factoring, so the quadratic formula is the best method.

Remember that the solution to the quadratic equation:   

                                                            is given by the formula:

In the equation:               

Factor out the common factor of 2 in the numerator, and reduce the fraction.

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