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If the roots of the equation x^3 – ax^2 + bx – c = 0 - Roots -PS

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If the roots of the equation x^3 – ax^2 + bx – c = 0 - Roots -PS  [#permalink]

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New post 15 Aug 2018, 06:59
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If the roots of the equation x^3 – ax^2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

(A) - 1/√3
(B) - 1
(C) 0
(D) 1
(E) 1/√3

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If the roots of the equation x^3 – ax^2 + bx – c = 0 - Roots -PS  [#permalink]

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New post 15 Aug 2018, 07:36
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Harshgmat wrote:
If the roots of the equation x^3 – ax^2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

(A) - 1/√3
(B) - 1
(C) 0
(D) 1
(E) 1/√3


OA:B

General cubic equation with roots \(p,q\) and \(r\) will be given by \((x-p)(x-q)(x-r)=0\).

\((x-p)(x-q)(x-r) =(x^2-(p+q)x+pq)(x-r)=x^3 -(p+q)x^2 +pqx -rx^2 +r(p+q)x-pqr)=x^3-(p+q+r)x^2+(pq+qr+rp)x-pqr\)

Equating above expression to \(x^3 – ax^2 + bx – c\), we get

\(b= (pq+qr+rp)\)

let \(3\) consecutive number be \(n-1,n,n+1\)

\(b = (n-1)n+n(n+1)+(n-1)(n+1) = 3n^2-1\)

\(n^2≥0\), minimum value of \(n^2\) is \(0\), leading to minimum value of \(b= -1\)
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Re: If the roots of the equation x^3 – ax^2 + bx – c = 0 - Roots -PS  [#permalink]

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New post 15 Aug 2018, 07:40
Harshgmat wrote:
If the roots of the equation x^3 – ax^2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

(A) - 1/√3
(B) - 1
(C) 0
(D) 1
(E) 1/√3



It's a cubic equation..
Let the roots be w,y,z
1) sum of all roots =w+y+z= a
2) sum of two roots at one time =wy+wz+yz=b
3) product of roots=wyz=c
So all 3 consecutive numbers as roots will give sum as the least when one is 0..
The other two -1 and 1, so roots are -1,0,1
And b=-1*1+0*1+0*-1=-1

B
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: If the roots of the equation x^3 – ax^2 + bx – c = 0 - Roots -PS &nbs [#permalink] 15 Aug 2018, 07:40
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