hectictar

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Usernamehectictar
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 #2
avatar+9460 
+1

I think it is the opposite guest... IDK the best way to explain it, but look at this graph:

 

https://www.desmos.com/calculator/din2ukxbds

 

We wish to find the max value of A while keeping S and R positive.

That appears to occur when  S = 0  and  R = 8/pi

Oct 30, 2020
 #1
avatar+9460 
+2

6j2 - 6j - 12

                          Factor  6  out of all the terms

=   6(j2 - j - 2)    (Notice here that if we distributed the 6, we would get back the previous expression.)

 

Now we want to force there to be a perfect square trinomial within the parenthesees. To do that, let's add and subtract half the coefficient of the middle term squared. That is, let's add and subtract  (1/2 * -1)2  which is  (-1/2)2  which is  1/4 . This will create a perfect square trinomial without changing the value of the expression.

 

=   6(j2 - j + \(\frac14\) - \(\frac14\) - 2)

                                       Now there is a perfect square trinomial which is highlighted below:

=   6(j2 - j + \(\frac14\) - \(\frac14\) - 2)

                                       Then  j2 - j + 1/4  factors as  (j - 1/2)2

=   6( (j - \(\frac12\))2  - \(\frac14\) - 2)

                                       And  2 = 8/4

=   6( (j - \(\frac12\))2  - \(\frac14\) - \(\frac84\))

                                       And -1/4 - 8/4  =  -9/4

=   6( (j - \(\frac12\))2  - \(\frac94\) )

                                       Distribute  6  to the terms in parenthesees

=   6( j - \(\frac12\) )2 -  6( \(\frac94\) )

 

=   6( j - \(\frac12\) )2\(\frac{27}{2}\)

 

Now it is in the form  c(j + p)2 + q,  where   c = 6,  p =  -\(\frac12\),  and  q = -\(\frac{27}{2}\)

 

So  q / p   =   ( -\(\frac12\) ) / ( -\(\frac{27}{2}\) )   =   ( -\(\frac12\) ) * ( -\(\frac{2}{27}\) )   =   \(\frac{1}{27}\)

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Oct 25, 2020