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(x - 12)(x + M) = M - 51

Simplify

x^2  - 12x + Mx  - 12M  = M - 51

x^2  + ( M - 12)x  - 13M +  51  =  0

For this to have only one solution, the disriminant  must  =  0

So  we have that

(M - 12)^2  - 4 (1) (51 - 13M)  =  0     simplify

M^2  - 24M + 144  - 204 + 52M  = 0

M^2 + 28M - 60  =  0     factor as

(M + 30) (M - 2)  =  0

Set each factor to 0  and solve for M  and we have that

M = -30   or  M  = 2

So.....M  = 2