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Let f(x) = 3x^2-3 and g(f(x)) = x^2 + x +1. Find the sum of all possible values of g(25).

 Nov 27, 2020
 #1
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f(25) = 1872

 

g(f(25)) = 1872^2 + 1872+1 = 3506257

 Nov 27, 2020
 #2
avatar+9198 
+1

Let's find what values of  x  make  f(x) = 25

 

f(x)  =  25

 

3x2 - 3  =  25

 

3x2  =  28

 

x2  =  28/3

 

x   =  ±√[ 28/3 ]

 

Now let's plug in both of these into  g(f(x))

 

g( 25 )   =   g(f( √[ 28/3 ] )   =   ( √[ 28/3 ] )2 + √[ 28/3 ] + 1   =   31/3 + √[ 28/3 ]

 

g( 25 )   =   g(f( -√[ 28/3 ] )   =   ( -√[ 28/3 ] )2 - √[ 28/3 ] + 1   =   31/3 - √[ 28/3 ]

 

And adding these two together we get...

 

(  31/3 + √[ 28/3 ]  )   +   (  31/3 - √[ 28/3 ]  )   =   31/3 + 31/3   =   62/3

 Nov 27, 2020
 #3
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+1

Agree with Hec        ....guest answer is incorrect

 

25 = 3x^2-3

0 = 3x^2-28      quadratic formula shows x = +- 2 sqrt21 / 3

 

Sub into g(x) :    (+2sqrt21/3)2 + 2 sqrt 21/3 +1 = ...........

                and     (-2sqrt21/3) - 2sqrt21/3 +1 = .............    you can do the math from here...should get same as hecticar

 Nov 27, 2020

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