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What is the sum of all possible values of $k$ for which $x^2 + kx - 12x + 16$ is the square of a binomial?

 Jan 13, 2018
 #1
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x^2 + kx - 12x  + 16    we can write

 

x^2  + ( k - 12)x  +   16

 

If this can be expressed as the square of a binomial....it will have only one root

 

Thus....the discriminant will  = 0  ....so.....

 

(k - 12)^2  -  4(16)   =  0

 

(k - 12)^2  -  64  = 0

 

(k - 12)^2  = 64       taking both roots, we have that

 

k - 12   = ±√64      so  either

 

k - 12  =  8                                 or             k - 12  =  -8

 

                   add  12 to both sides

 

k  = 20                or                  k  = 4

 

 

Check

 

 

x^2  +  8x  + 16   factors as   ( x + 4)^2

 

And

 

x^2 - 8x  +  16   factors as  (x - 4)^2

 

 

cool cool cool

 Jan 13, 2018

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