Question 1146943
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The equation

    2*(x-3)*(x-5) = k

is equivalent to

    2x^2 - 16x + 30 = k,   or

    2x^2 - 16x + (30-k) = 0,  or

     x^2 - 8x + {{{(30-k)/2}}} = 0.


According to Vieta's theorem, the sum of the roots is equal to coefficient at "x" with the opposite sign, i.e. 8.


The difference of the roots is equal to 6 (given).


If "a" is the greater root, then the other root is (8-a), and the difference between them is


    6 = a - (8-a) = 2a - 8,

which implies  

    2a = 6 + 8 = 14,

     a = 7.


Thus the greater root is  a= 7, while the smaller root is  8-a = 8-7 = 1.


The product of the roots, 7*1, is equal to the constant term  {{{(30-k)/2}}}, according to Vieta's theorem (again (!) ).

Thus you have this equation for k


    {{{(30-k)/2}}} = 7,

which implies

    30 - k = 14.


Hence,  k = 30-14 = 16.
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Solved.


VERY GOOD problem on Vieta's theorem (!)