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

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Math Expert
Joined: 02 Sep 2009
Posts: 59721
Let a, b, and c be three distinct one-digit numbers. What is the maxim  [#permalink]

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25 Apr 2019, 03:12
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Difficulty:

55% (hard)

Question Stats:

59% (02:30) correct 41% (02:15) wrong based on 39 sessions

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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

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Re: Let a, b, and c be three distinct one-digit numbers. What is the maxim  [#permalink]

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25 Apr 2019, 10:10
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

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Joined: 12 Sep 2015
Posts: 4150
Re: Let a, b, and c be three distinct one-digit numbers. What is the maxim  [#permalink]

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28 Nov 2019, 11:59
1
Top Contributor
Bunuel wrote:
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

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

Cheers,
Brent
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Re: Let a, b, and c be three distinct one-digit numbers. What is the maxim   [#permalink] 28 Nov 2019, 11:59
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