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Board of Directors D
Joined: 01 Sep 2010
Posts: 3417
If x^2 = 2^x, what is the value of x ?  [#permalink]

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Top Contributor
33 00:00

Difficulty:   65% (hard)

Question Stats: 62% (02:13) correct 38% (02:16) wrong based on 804 sessions

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If $$x^2 = 2^x$$, what is the value of x ?

(1) $$2x = (\frac{x}{2})^3$$

(2) $$x = 2^{x-2}$$

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Math Expert V
Joined: 02 Aug 2009
Posts: 8281
If x^2 = 2^x, what is the value of x ?  [#permalink]

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6
5
carcass wrote:
If $$x^2 = 2^x$$, what is the value of x ?

(1) $$2x = (\frac{x}{2})^3$$

(2) $$x = 2^x-2$$

Hi..

If $$x^2 = 2^x$$, what is the value of x ?

(1) $$2x = (\frac{x}{2})^3$$
$$x*(\frac{x^2}{8}-2)=0$$
we get x as 0, 4 or -4..
when we substitute value in $$x^2 = 2^x$$, ONLY x = 4 is possible..
sufficient

(2) $$x = 2^{x-2}$$
or $$x=2^{x-2}$$...
2^x has to be multiple of 2 and so also x should be in power of 2 that is 2,4,8,16...AND x will fit in only at x=4 as $$2^{4-2}$$
ONLY when x is 4 both sides are 4.
.
suff

D

carcass i believe statement II should be $$x=2^{x-2}$$.. here possible value is 4 SAME as statement I.. pl relook into Q
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##### General Discussion
Board of Directors D
Joined: 01 Sep 2010
Posts: 3417
Re: If x^2 = 2^x, what is the value of x ?  [#permalink]

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2
Top Contributor
Hi Chetan,

fixed, was a question of formatting rather than a typo.

Thank you so much for pointing out
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Intern  B
Joined: 06 May 2017
Posts: 4
Re: If x^2 = 2^x, what is the value of x ?  [#permalink]

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Statement II can be turned into 4x=2^x which clearly has the only answer x=4, so it is sufficient
Manager  G
Joined: 14 Oct 2012
Posts: 158
Re: If x^2 = 2^x, what is the value of x ?  [#permalink]

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2
My 2 cents:
Attachment: my 2 cents.png [ 3.04 MiB | Viewed 17800 times ]
Intern  B
Joined: 28 Aug 2016
Posts: 6
If x^2 = 2^x, what is the value of x ?  [#permalink]

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hey,
Still confused on statement 2
x=2^x-2
plugged that into x^2=2^x
2^(2x)-2^(2)=(2^x-2^2)
2x-2=2x-2
where I'm going wrong
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: If x^2 = 2^x, what is the value of x ?  [#permalink]

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3
2
cocojatti92 wrote:
If $$x^2 = 2^x$$, what is the value of x ?

(1) $$2x = (\frac{x}{2})^3$$

(2) $$x = 2^{x-2}$$

hey,
Still confused on statement 2
x=2^x-2
plugged that into x^2=2^x
2^(2x)-2^(2)=(2^x-2^2)
2x-2=2x-2
where I'm going wrong

First of all, notice that (2) is $$x = 2^{x-2}$$ (2 in power of x-2) not $$x = 2^x-2$$.

$$x = 2^{x-2}$$;

$$x = \frac{2^x}{4}$$;

Since $$x^2 = 2^x$$, then $$x = \frac{x^2}{4}$$;

$$x(\frac{x}{4}-1)=0$$;

x = 0 or x = 4.

Only x = 4 satisfies $$x^2 = 2^x$$, this x = 4. Sufficient.

Hope it's clear.
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Senior Manager  P
Joined: 09 Jun 2014
Posts: 353
Location: India
Concentration: General Management, Operations
Re: If x^2 = 2^x, what is the value of x ?  [#permalink]

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chetan2u wrote:
carcass wrote:
If $$x^2 = 2^x$$, what is the value of x ?

(1) $$2x = (\frac{x}{2})^3$$

(2) $$x = 2^x-2$$

Hi..

If $$x^2 = 2^x$$, what is the value of x ?

(1) $$2x = (\frac{x}{2})^3$$
$$x*(\frac{x^2}{8}-2)=0$$
we get x as 0, 4 or -4..

when we substitute value in $$x^2 = 2^x$$, ONLY x = 4 is possible..
sufficient

(2) $$x = 2^x-2$$
or $$x+2=2^x$$...
2^x has to be multiple of 2 and so also x+2 should be in power of 2 that is 2,4,8,16...AND x+2 will fit in only at x+2=4 as 2^2
ONLY when x is 2 both sides are 4.
.
suff

although ans is D, but values of x differ as 4 in statement I and 2 in II
generally not possible in OG..

carcass i believe statement II should be $$x=2^{x-2}$$.. here possible value is 4 SAME as statement I.. pl relook into Q

That was a very helpful solution!!

Chetan,

I have a query. I was able to solve the eqautions and get the values but I didnt plug in back and check that in the main equation.So I chose E

Can you please explain when and what pattern do you see and keep a check a to plugin back the multiple values.

For example in modules questions we do...do you also check for plolynomial degree equations..

Please tell your line of thinking/approach when attempting this variety of questions!!
Intern  B
Joined: 28 Apr 2019
Posts: 30
Re: If x^2 = 2^x, what is the value of x ?  [#permalink]

### Show Tags Re: If x^2 = 2^x, what is the value of x ?   [#permalink] 25 Oct 2019, 23:50
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# If x^2 = 2^x, what is the value of x ?  