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The quadratic 4/3x^2+4x+1 can be written in the form a(x+b)^2+c, where a, b, and c are constants. What is abc? Give your answer in simplest form.

 Jun 27, 2018
 #1
avatar+98182 
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(4/3)x^2  + 4x + 1       factor out the 4/3

 

(4/3)  [ x^2  + 3x   +  3/4 ]       complete the square  on the quadratic inside the brackets

 

(4/3)  [ x^2  + 3x + 9/4  + 3/4  - 9/4 ]     factor the first three terms and simplify the rest

 

(4/3) [ (x + 3/2)^2  - 3/2 ]     distribute the (4/3)  over the terms

 

(4/3) ( x + 3/2)^2    - 2    =   (4/3) ( x + 3/2)^2  + (-2) 

 

So...in the form   a( x +b)^2 + c

 

a = (4/3)   b = (3/2)   and  c  = -2

 

So

 

abc  =  ( 4/3)(3/2)(-2)   =  -24 / 6   =   -4

 

 

cool cool cool

 Jun 27, 2018

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