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655-705 (Hard)|   Algebra|   Exponents|      
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DisciplinedPrep
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo Updated on: 02 Dec 2019, 04:49
Originally posted by DisciplinedPrep on 02 Dec 2019, 04:13.
Last edited by Bunuel on 02 Dec 2019, 04:49, edited 1 time in total.
Renamed the topic and edited the question.
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65% (hard)

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59% (01:50) correct 41% (02:30) wrong based on 247 sessions

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If \((3 + 2√2)^{(x^2 - 3)} + (3 - 2√2)^{(x^2 - 3)} = b\), then which of the following could be a value of b?

A. 0
B. 2
C. √2
D. -√2
E. 2√2
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B
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ManjariMishra
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 02 Dec 2019, 04:42
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can anybody please explain how to approach this ?
chetan2u, Bunuel, VeritasKarishma, Gladiator59, generis
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GMAT Focus 1: 735 Q90 V89 DI81
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 02 Dec 2019, 06:20
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DisciplinedPrep
If \((3 + 2√2)^{(x^2 - 3)} + (3 - 2√2)^{(x^2 - 3)} = b\), then which of the following could be a value of b?

A. 0
B. 2
C. √2
D. -√2
E. 2√2
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Hi ManjariMishra
The quickest thing I can think of.
What happens when the power is 0, the term becomes 1..
So when x^2=3, both terms are 1 and answer will be 1+1=2
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Pooja132
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 02 Dec 2019, 07:11
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any other method to solve this ?
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 02 Dec 2019, 19:36
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Official Solution

\((3 + 2√2)^{(x^2 - 3)}\) be y.
(3 + 2√2) and (3 - 2√2) are conjugate numbers.
Since they are conjugate numbers, (3 + 2√2) * (3 - 2√2) = 1
So, (3 + 2√2) = \(\frac{1}{(3−2√2)}\)
or (3 - 2√2) = \(\frac{1}{(3+2√2)}\)
Now, \((3 - 2√2)^{(x^2 - 3)}\) = 1/\((3 + 2√2)^{(x^2 - 3)}\) = \(\frac{1}{y}\)

Equation can be written as
\((3 + 2√2)^{(x^2 - 3)} + (3 - 2√2)^{(x^2 - 3)}\) = y + \(\frac{1}{y}\) = b
As a rule, the expression: y + \(\frac{1}{y }\)≥ 2 or y + \(\frac{1}{y}\) ≤ -2
From the options, it is clear that y + \(\frac{1}{y}\) can take the value 2.

So, b can take the value 2. Hence the answer is "2"
Therefore, the correct answer is option B. 2

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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 02 Dec 2019, 21:18
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ManjariMishra
can anybody please explain how to approach this ?
chetan2u, Bunuel, VeritasKarishma, Gladiator59, generis
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I agree with chetan2u.

When I looked at the equation, I wanted to simplify it as much as possible since options give very simple values of b.

So it looks something like this:

\((p + q)^r + (p - q)^r \)

Now, if r = 0 or 1, it will simply the expression because the q's will get cancelled.

Putting r = 0 (which means x = sqrt(3)), we get 1 + 1 = b = 2 (which is in the options)
So that is the answer.

If it were not, I would have put r = 1 (which means x = 2). We would have got 3 + 3 = b = 6
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo Updated on: 02 Dec 2019, 23:34
Originally posted by nick1816 on 02 Dec 2019, 21:40.
Last edited by nick1816 on 02 Dec 2019, 23:34, edited 1 time in total.
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As y>0 (for all real values of x) in this question, (y+1/y) can't be negative.

Also, B and E both are possible answers. Please edit one of them.

DisciplinedPrep
Official Solution

\((3 + 2√2)^{(x^2 - 3)}\) be y.
(3 + 2√2) and (3 - 2√2) are conjugate numbers.
Since they are conjugate numbers, (3 + 2√2) * (3 - 2√2) = 1
So, (3 + 2√2) = \(\frac{1}{(3−2√2)}\)
or (3 - 2√2) = \(\frac{1}{(3+2√2)}\)
Now, \((3 - 2√2)^{(x^2 - 3)}\) = 1/\((3 + 2√2)^{(x^2 - 3)}\) = \(\frac{1}{y}\)

Equation can be written as
\((3 + 2√2)^{(x^2 - 3)} + (3 - 2√2)^{(x^2 - 3)}\) = y + \(\frac{1}{y}\) = b
As a rule, the expression: y + \(\frac{1}{y }\)≥ 2 or y + \(\frac{1}{y}\) ≤ -2
From the options, it is clear that y + \(\frac{1}{y}\) can take the value 2.

So, b can take the value 2. Hence the answer is "2"
Therefore, the correct answer is option B. 2

ManjariMishra Pooja132
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 02 Dec 2019, 22:54
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DisciplinedPrep
If \((3 + 2√2)^{(x^2 - 3)} + (3 - 2√2)^{(x^2 - 3)} = b\), then which of the following could be a value of b?

A. 0
B. 2
C. √2
D. -√2
E. 2√2
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Let x = \sqrt{3}
x^2 = 3

(3 + 2√2)^0 + (3 - 2√2)^0 = b
1+1 = b

b could be 2

B is correct
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 01 Jun 2020, 13:09
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DisciplinedPrep
Official Solution

\((3 + 2√2)^{(x^2 - 3)}\) be y.
(3 + 2√2) and (3 - 2√2) are conjugate numbers.
Since they are conjugate numbers, (3 + 2√2) * (3 - 2√2) = 1
So, (3 + 2√2) = \(\frac{1}{(3−2√2)}\)
or (3 - 2√2) = \(\frac{1}{(3+2√2)}\)
Now, \((3 - 2√2)^{(x^2 - 3)}\) = 1/\((3 + 2√2)^{(x^2 - 3)}\) = \(\frac{1}{y}\)

Equation can be written as
\((3 + 2√2)^{(x^2 - 3)} + (3 - 2√2)^{(x^2 - 3)}\) = y + \(\frac{1}{y}\) = b
As a rule, the expression: y + \(\frac{1}{y }\)≥ 2 or y + \(\frac{1}{y}\) ≤ -2
From the options, it is clear that y + \(\frac{1}{y}\) can take the value 2.

So, b can take the value 2. Hence the answer is "2"
Therefore, the correct answer is option B. 2

ManjariMishra Pooja132
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Hello sir,

could you please explain how you managed: y + \(\frac{1}{y }\)≥ 2 or y + \(\frac{1}{y}\) ≤ -2
Also, I couldn't understand " y + \(\frac{1}{y}\) can take the value 2"

Thanks
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If (3 + 2√2)^(x^2 - 3) + (3 - 2√2)^(x^2 - 3) = b, then which of the fo 13 Aug 2025, 11:08
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