Help plz!!

Question

A unary operator ◊ is defined as ◊ x = x2 – 4x + 3. If ◊β = ◊(3 – 2β), what is the value of β? A) 1 only B) – 1 only C) ± 1 D) None of these

Matt@VeritasPrep · Answer

Let's change the diamond to a # so it's easier to type.

#x, for any value of x, is defined as x² - 4x + 3. So #2 = 2² - 4*2 + 3, #10 = 10² - 4*10 + 3, etc.

From there, we can say that

#ß = ß² - 4ß + 3

and

#(3 - 2ß) = (3 - 2ß)² - 4*(3 - 2ß) + 3

Since we're told these are equal, we've got

ß² - 4ß + 3 = (3 - 2ß)² - 4*(3 - 2ß) + 3

or

ß² - 4ß + 3 = 9 - 12ß + 4ß² - 12 + 8ß + 3

or

0 = 3ß² - 3

or

3 = 3ß²

or

1 = ß²

so ß = ±1

Matt@VeritasPrep · Answer

You could also cheat a bit here if you have no time or no idea what to do. Since the function tells us that #ß = #(3 - 2ß), it's pretty reasonable to guess that ß = 3 - 2ß, so ß = 1 is probably a solution.

Since we're squaring terms inside the function, there's also a decent chance that -1 is a solution too, so 1 and -1 seem like good guesses.

Matt@VeritasPrep · Answer

We could also save time by trying the answers. Since all of them (except D) involve 1 and -1, let's try both.

First case, ß = 1:

#1 = #(3 - 2*1)

#1 = #1

Well, duh!

Second case, ß = -1:

#(-1) = #(3 - 2*(-1))

#(-1) = #5

(-1)² - 4*(-1) + 3 = 5² - 4*5 + 3

1 + 4 = 5² - 20

Also true! So ß = -1 is another solution.

Matt@VeritasPrep · Answer

It's also reasonable to wonder how we can exclude D, since we could have lots of solutions, right?

When we go from this step

#(3 - 2ß) = (3 - 2ß)² - 4*(3 - 2ß) + 3

to this step

ß² - 4ß + 3 = (3 - 2ß)² - 4*(3 - 2ß) + 3

We can see that our equation is in one variable (ß) and its highest power is ². So we invoke the Fundamental Theorem of Algebra - which is as mighty as it sounds! - to tell us that there are AT MOST two solutions to this problem.

Jay@ManhattanReview · Answer

Hmna wrote:A unary operator ◊ is defined as ◊ x = x2
– 4x + 3. If ◊β = ◊(3 – 2β), what is the value of β?
A) 1 only
B) – 1 only
C) ± 1
D) None of these

Hi Hmna,

This is not a GMAT question. It helps to prepare GMAT type questions. Pl. post only GMAT questions.

We have ◊ x = x^2– 4x + 3;

We are given that ◊β = ◊(3 – 2β)

◊B = B^2 - 4B + 3;

◊(3 – 2B) = (3 – 2B)^2 - 4(3 – 2B) + 3 = 9 - 12B + 4B^2 - 12 + 8B + 3 = 4B^2 - 4B

=> B^2 - 4B + 3 = 4B^2 - 4B

=> 3B^2 = 3

=> B^2 = 1

=> [spoiler]B = +/-1[/spoiler]

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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Matt@VeritasPrep · Answer

Jay@ManhattanReview wrote:
This is not a GMAT question.

It does test GMAT concepts, though, so I'd say it's fair game and not bad practice. Its only non-GMATy features are the use of the term 'unary' and the four answer choices.

Jay@ManhattanReview · Answer

Matt@VeritasPrep wrote:Jay@ManhattanReview wrote: This is not a GMAT question. It does test GMAT concepts, though, so I'd say it's fair game and not bad practice. Its only non-GMATy features are the use of the term 'unary' and the four answer choices. I agree with you, Matt; however, we must advise the new poster that the source being referred to is probably not the GMAT source. And maybe he/she is practicing few non-GMAT type questions. As I see that Hmna has posted four questions and each have only four options.

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