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# If the equation x^3 - ax^2 +bx -a = 0 has three real roots

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Manager
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If the equation x^3 - ax^2 +bx -a = 0 has three real roots [#permalink]

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08 Nov 2009, 14:02
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85% (hard)

Question Stats:

45% (01:37) correct 55% (01:40) wrong based on 218 sessions

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If the equation x^3 - ax^2 +bx -a = 0 has three real roots then the following id true

A. a=11
B. a not equal to 1
C. b = 1
D. b not equal to 1

How to solve equations with power 3?
Any specific tips to handle questions with such equations?
[Reveal] Spoiler: OA

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08 Nov 2009, 16:05
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papillon86 wrote:
If the equation x^3 - ax^2 +bx -a = 0 has three real roots then the following id true

a) a=11
b) a not equal to 1
c) b = 1
d) b not equal to 1

How to solve equations with power 3?
Any specific tips to handle questions with such equations?

I don't think we need to know how to solve cubic equation for GMAT.

For above question:

$$x^3 - ax^2 +bx -a = 0$$

$$x^2*(x-a)+(bx-a)=0$$

We are told that this equation has THREE roots. If look closer we notice that when $$b=1$$ we can write the equation as:

$$x^2*(x-a)+(x-a)=0$$ --> $$(x^2+1)*(x-a)=0$$ and this equation has only ONE root $$x=a$$.

Hence given equation to have three roots b must not equal to 1.

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11 May 2011, 23:23
assuming b= 1

x^2(x-a)+ 1(x-a) = (x^2+1) * (x-a)

x^2 != -1 meaning b != 1.

Hence D.
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Manager
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Re: If the equation x^3 - ax^2 +bx -a = 0 has three real roots [#permalink]

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27 Nov 2013, 01:50
Good question.

The point here is to reduce the equation to x2(x-a)+bx-a=0.

Now if b=1, the equation becomes (x2+1)(x-a) = 0

The above equation has one real and two imaginary roots and hence does not satisfy the condition given in the quesiton(3 real roots)

Therefore b cannot be equal to 1.

Thanks.
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Re: If the equation x^3 - ax^2 +bx -a = 0 has three real roots [#permalink]

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19 Jun 2015, 08:08
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Re: If the equation x^3 - ax^2 +bx -a = 0 has three real roots [#permalink]

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19 Jun 2015, 17:52
Hi All,

While this is an old series of posts, it's worth noting that this question is not written in proper GMAT 'style' and is not something that you'll be likely to see on Test Day. In the event that you did see this type of equation, the likely 'first step' would be to factor out an X (with the likely result that you would be left with a standard Quadratic of some kind). THAT type of Algebra IS tested on the GMAT, so you should be sure to put in the necessary work to build up your basic Algebra skills and pattern-matching abilities.

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Rich
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Re: If the equation x^3 - ax^2 +bx -a = 0 has three real roots   [#permalink] 19 Jun 2015, 17:52
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