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25 Apr 2019, 03:12
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
61% (02:17) correct
39%
(02:32)
wrong
based on 195
sessions
History
Date
Time
Result
Not Attempted Yet
Let a, b, and c be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation \((x-a)(x-b)+(x-b)(x-c)=0\) ?
(A) 15
(B) 15.5
(C) 16
(D) 16.5
(E) 17
(A) 15
(B) 15.5
(C) 16
(D) 16.5
(E) 17
25 Apr 2019, 10:10
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Let us solve the equation, \((x−a)(x−b)+(x−b)(x−c)=0\)
\(x^2-bx-ax+ab+x^2-cx-bx+cb = 0\)
\(2x^2-x(a+2b+c)+ab+cb = 0\)
Sum of the roots of the equation = -b/a
a = 2
b = -(a+2b+c)
As a, b and c are distinct single digit integers, let take b as 9, a as 8 and c as 7. (As we need to maximize the value )
Sum = (18+15)/2 = 16.5
D is the answer.
\(x^2-bx-ax+ab+x^2-cx-bx+cb = 0\)
\(2x^2-x(a+2b+c)+ab+cb = 0\)
Sum of the roots of the equation = -b/a
a = 2
b = -(a+2b+c)
As a, b and c are distinct single digit integers, let take b as 9, a as 8 and c as 7. (As we need to maximize the value )
Sum = (18+15)/2 = 16.5
D is the answer.
General Discussion
28 Nov 2019, 11:59
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Bunuel
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
