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The operation a # b is defined by a + 2b + 3ab.  Solve the equation (a # a) # a = a # (a # a).

 May 18, 2020
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a # b  =  a + 2b + 3ab

 

For (a # a ) # a:

 

Working inside the parentheses first:

     a # a  =

    The second a takes the place of b in the formula:  a # b  =  a + 2b + 3ab

    so we have:  a # a  =  a + 2a + 3aa  =  3a + 3a2

 

Now we have:  (3a + 3a2) # a

    In this case, the expression "3a + 3a2" takes the place of the a

                        while a takes the place of b

   so we have:  a # a  =  (3a + 3a2) + 2a + 3(3a + 3a2)a

                                  =  3a + 3a2 + 2a + 9a2 + 9a3

                                  =  5a + 13a2 + 9a3

 

This is the left-hand side. Can you do the right-hand side?

If you can, set the two sides equal to each other and solve ...

 May 18, 2020

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