Let a, b, and c be three distinct one-digit numbers. What is the maxim

GIVEN: \((x-a)(x-b)+(x-b)(x-c)=0\)

Rewrite as: \((x-b)[(x-a)+(x-c)]=0\)

Simplify: \((x-b)[2x-a-c]=0\)

Rewrite as: \((x-b)[2x-(a+c)]=0\)

So, EITHER \(x-b=0\) OR \(2x-(a+c)=0\)

If \(x-b=0\), then the greatest possible value of x occurs when b = 9
When b = 9, one solution (root) is x = 9

If \(2x-(a+c)=0\), then the greatest possible value of x occurs when the sum (a+c) is maximized
However, since a b and c are distinct integers (and since we have already let b = 9), the greatest possible value of (a+c) occurs when a = 7 and c = 8
In this case we get: \(2x-(7+8)=0\)
Simplify: \(2x-15=0\)
So, when a = 7 and c = 8, another possible solution (root) is x = 7.5

So the maximum value of the sum of the roots equals = 9 + 7.5
= 16.5

Answer: D

Cheers,
Brent