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# If f(x^2) = x^4–6x^2+5, which of the following must be true?

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Director
Joined: 18 Feb 2019
Posts: 582
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GMAT 1: 460 Q42 V13
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If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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24 Mar 2019, 00:31
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Difficulty:

65% (hard)

Question Stats:

58% (02:50) correct 42% (02:03) wrong based on 36 sessions

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If f(x^2) = x^4–6x^2+5, which of the following must be true?

A. f(x^2)=f(1+x^2)
B. f(x^2)=f(2-x^2)
C. f(x^2)=f(3-x^2)
D. f(x^2)=f(5-x^2)
E. f(x^2)=f(6-x^2)
Director
Joined: 28 Jul 2016
Posts: 609
Location: India
Concentration: Finance, Human Resources
GPA: 3.97
WE: Project Management (Investment Banking)
Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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24 Mar 2019, 07:04
I tried usiing conventional approach,
Substitute the value of $$x^2$$ to each value given in option from A to E.
Only E satisfies the given equation
Is their a shorter way to do this.
also is possible to write question int he mathematical formula format.
It would be easier to read
Intern
Joined: 24 Oct 2018
Posts: 3
Location: India
Schools: ISB '21 (S)
GMAT 1: 710 Q49 V37
Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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25 Mar 2019, 02:53
kiran120680 wrote:
If f(x^2) = x^4–6x^2+5, which of the following must be true?

A. f(x^2)=f(1+x^2)
B. f(x^2)=f(2-x^2)
C. f(x^2)=f(3-x^2)
D. f(x^2)=f(5-x^2)
E. f(x^2)=f(6-x^2)

You can factorize the above eqn to (x^2-5)(x^2-1)..then it is easy
Manager
Joined: 27 Oct 2017
Posts: 72
Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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07 Apr 2019, 10:46
Can someone please explain it in detail?

Posted from my mobile device
Director
Joined: 28 Jul 2016
Posts: 609
Location: India
Concentration: Finance, Human Resources
GPA: 3.97
WE: Project Management (Investment Banking)
Re: If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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07 Apr 2019, 12:12
1
arorni wrote:
Can someone please explain it in detail?

Posted from my mobile device

Let me try
$$f(x^2) = x^4–6x^2+5$$, which of the following must be true?

Lets factor the given equation first
$$x^4–6x^2+5$$ = $$(x^2-5)(x^2-1)$$

A.$$f(x^2)=f(1+x^2)$$
B.$$f(x^2)=f(2-x^2)$$
C. $$f(x^2)=f(3-x^2)$$
D. $$f(x^2)=f(5-x^2)$$
E. $$f(x^2)=f(6-x^2)$$

Now for each options put the given values of f(n) instead of f(x^2) in the factor
$$(x^2-5)(x^2-1)$$
Such that for example if you put
$$(1+x^2)$$ instead of $$x^2$$ in the function value $$(x^2-5)(x^2-1)$$ . It will become
=$$(1+x^2-5)(1+x^2-1)$$=$$(+x^2-4)(x^2)$$. This is not equal to original expression.

You will find only E fits in substitute$$(6-x^2)$$ in place of$$x^2$$

$$(x^2-5)(x^2-1) = (6-x^2-5)(6-x^2 -1)= (x^2-1)(x^2-5)$$.
This is equal to original expression. Hence (E)
Intern
Joined: 07 Apr 2019
Posts: 3
If f(x^2) = x^4–6x^2+5, which of the following must be true?  [#permalink]

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07 Apr 2019, 13:35
1
kiran120680 wrote:
If f(x^2) = x^4–6x^2+5, which of the following must be true?

A. f(x^2)=f(1+x^2)
B. f(x^2)=f(2-x^2)
C. f(x^2)=f(3-x^2)
D. f(x^2)=f(5-x^2)
E. f(x^2)=f(6-x^2)

arorni wrote:
Can someone please explain it in detail?

You can reduce the difficulty of the question once you realize that the function is quadratic and not of 4th degree.

Setting $$z = x^2$$, one can see that the function $$f$$ itself is $$f(z) = z^2 -6z +5$$, and the question is: "For which $$\beta$$ is $$f(\beta -z) = f(z)$$ true for all $$z \geq 0$$?" (modify it for answer choice A). In particular, the relation is true for $$z = 0$$ which boils down to $$f(\beta) = f(0)$$ or $$\beta^2 -6\beta +5 = 5$$, and therefore $$\beta = 6$$.

In practice, you simply calculate $$f(1), f(2), f(3), f(5)$$, and $$f(6)$$ and look which one returns the value 5.
If f(x^2) = x^4–6x^2+5, which of the following must be true?   [#permalink] 07 Apr 2019, 13:35
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