10.    Solve the quadratic equation:   .

              Solution:   

                 Of course the easiest way to solve a quadratic equation is to factor it. 

                 But how do you factor a trinomial with numbers like this? 

                                   Find two numbers whose product is 47816, that add up to ?  Lots of luck guessing the right combination of numbers!!

                                   Of course, if you have a graphing calculator you can use a program on the TI83+ or TI84 called “Polysmlt”, or with any graphing calculator, you can sketch the graph of   and find the zeros (or x-intercepts) of this graph.  These will be the solutions to the equation.  (The trick in graphing this will be to determine the best window to view the intercepts!  Unless you are really good with the graphing calculator, I don't recommend this!) 

                                   Using algebra techniques, you could always solve using the quadratic formula, but in this case, since the coefficient of is , you could also use the completing the square method.  When completing the square, your first step is to take the constant term to the right side, and place a blank space on each side to set up completing the square, like this:

                                 
Next, (assuming that you have , which you do have)

                                  take half of the coefficient and square it.  

                                  Half of  is , and .  Add this to each side:

The left side is automatically a perfect square trinomial,

                                  obtained either by taking half of the or the square root of .  Either way, you get :

 

                                     
Next take the square root of each side.  If    happens to be a perfect square, then the original problem could have been factored, but with the size of those numbers in the original equation, it's not likely you would have been able to do it by the factoring method.  Take your calculator and take the square root of , and it is indeed .

   or     
   or      
          or     

Since   turned out to be a perfect square, it means that the original problem could have been solved by factoring.  Now that you know what the solution is (a little “inside” information), could you actually solve the problem by factoring?  Let’s try again from the top. 

Since     and     are solutions, this means

that   and  are factors.  So the quadratic equation

             would have factored as

                                            or 

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