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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?

Guest Jan 13, 2018
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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

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

CPhill  Jan 13, 2018

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