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GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is
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Bunuel
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GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:00
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If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of x ?

A. 1/6
B. 1/5
C. 1/3
D. 2/3
E. 9/10


 


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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:16
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My answer is D).

Assume 3^x = A, then the equation becomes: 3A^2+4A-20=0 -> (A-2)(3A+10)=0 -> A=2 (3A+10>0).

So 3^x = 2. Looking at the choices: A)-C) can be quickly eliminated because say if x = 1/3, then 3^(1/3) = 2, 3 = 2^3, clearly not going to work.

E) can also be eliminated because by the same logic, 3^9 = 2^10. We know that 3^x will climb much faster than 2^x when x becomes larger.

D) works because if 3^(2/3) = 2, then 3^2 = 9 = 2^3 = 8, close enough. So D).
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:21
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Kudos
Let 3 ^x be t
equation will become 3t^2+4t-20=0
On solving this we get t as -10/3 and 2.Now t to be -10/3 x should be -ve not possible from options.
So, t=2 which means 3^x=2.Since 3^(1/2) is close to 1.71 so option should be slightly greater than 0.5, so option D IMO.
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:21
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\(3*3^{(2x)} + 4(3^x) – 20 = 0\)

Let:
\(a=3^x\)

\(3a^2 + 4a – 20 = 0\)
\((a-2)(3a+10)=0\)

As \(a=3^x\) must be positive -> \(a=2\) -> \(3^x = 2\) -> x is approximate to \(3^\frac{2}{3}\)

IMO D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:25
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Asked: If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of x ?

Let \(3^x = t\)

\(3*t^2 + 4t - 20 = 0\)
\(t = (-4 + - \sqrt{16 + 240}) / 6 = (-4 + - \sqrt{256})/6 = (-4+-16)/6 = {2, -10/3}\)

3^x = -10/3; NOT POSSIBLE
\(3^x = 2 \)
x = log 2/ log 3 = .6309 ~ 2/3

IMO D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:29
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Kudos
Let 3^x be k

3k^2+4k -20 = 0

k = (-4 +- root(16+240))/2.3

k = 12/6 or -20/6

k = 2

3^x = 2

x = 2/3
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:32
3^x can be considered as a variable and the resultant quadratic equation xan be solved.

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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:32
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Hello

Here is my solution.
The correct answer is D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:41
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Kudos
let 3^x = a. The equation can be written as:

3a^2 + 4a - 20 = 0
=> 3a^2 + 10a - 6a -20 = 0
we get a = 2
3^x = 2

square root of 3 = 1.732
so the power should be more than 1/2 (A,B,C eliminated). But the power should not be too far away from 1/2 since 1.732 is close to 2.
The closest power of 3 that will give the value 2 is 2/3 (9/10 >> 2/3). Hence D is the answer.
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:42
Bunuel wrote:
If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of x ?

A. 1/6
B. 1/5
C. 1/3
D. 2/3
E. 9/10


 


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3^x= y
3y^2 + 4y-20 =0

y=2 or y=-10.3 (it cannot be negatice therefore out )

3^x =2
=>x=1/6 approx since cube of 3 is 1.4 and square of 1.4 is 1.96 close to 2

Therefore IMO A
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:42
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taking \(3^x\) as z

\(3z^2 + 4z - 20 = 0\)

solving for z, \(z = 2\) (as other value as -ve)

therefore \(3^x = 2\)

or, \(x=\frac{\ln \left(2\right)}{\ln \left(3\right)}\)

Closest is 2/3 or 0.6

Option D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:48
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If we mark 3^x as a, then we can see that we need to deal with a simple quadratic equation.
My answer is D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 07:51
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\(3^{(2x+1)} + 4(3^x) – 20 = 0\)
expression can be simplified as
3^2x +3+4*3^x-20=0
3^2x+4*3^x-17=0
given values of answer option round off to nearest decimal
A. 1/6 ;.166
B. 1/5 ; .2
C. 1/3;.33
D. 2/3;.66
E. 9/10;.9
plugin values of the options we see that near ~ value comes are 2/3 ; .66 such that \(3^{(2x+1)} + 4(3^x) – 20 = 0\)

option D is correct


Bunuel wrote:
If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of x ?

A. 1/6
B. 1/5
C. 1/3
D. 2/3
E. 9/10


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:03
1
Kudos
Let us take y = 3^x
Hence, the equation changes to 3y^2 +4y-20 = 0

Finding the factors, (3y+10)(3y-6)
since the value has to be +ve, we have y =2

this implies, 3^x = 2
Hence, x = 2/3 , approx value of 3^(2/3) = 2.08

IMO : D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:06
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IMO OA should be D

\(3^(2x+1) + 4(3^x) – 20 = 0\)

=> \(3^(2x) * 3 + 4(3^x) – 20 = 0
\)
=> Let's say \(3^x = t\)

=> \(3t^2 + 4t - 20 = 0
\)

after solving the quadratic equation for t => t = 2 & \(\frac{- 10}{3}\)

Taking the positive value

\(t = 3^x = 2\)

\(3^1 = 3 \)

\(3^0.5 = 1.7\)

x should be in between 1 & 0.5. So, A, B & C are out. E is too close to 1. Reject E

x should be ~ 0.66 or \(\frac{2}{3}\)

Hence D option.
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:16
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Say: a = 3^x (a >0)
=> 3a^2 + 4a - 20 = 0
=> a = 2
=> 3^x = 2
approximate x = 2/3
=> IMO: D
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:18
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3^(2x+1)+4*(3^x)–20=0
=> 3*3^2x+4*3^x-20=0

Lets assume, 3^x=p

Therefore,

3p^2+4p-20=0
3p^2+10p-6p-20=0
(3p+10)(p-2)=0
p=2 or p=-10/3

3^x=2

x~2/3

0r

3^x=-.33 which is not possible becuase of negative sign

So D is the answer
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:23
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Bunuel wrote:
If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of x ?

A. 1/6
B. 1/5
C. 1/3
D. 2/3
E. 9/10


 


This question was provided by GMAT Club
for the GMAT Club Olympics Competition

Win over $40,000 in prizes such as Courses, Tests, Private Tutoring, and more

 



\(3^{(2x+1)} + 4(3^x) – 20 = 0\)
\(3.3^{(2x)} + 4(3^x) – 20 = 0\)
\(3. {(3^x)}^2 + 4(3^x) – 20 = 0\)
\( Let (3^x) = y \)
\( 3y^2 + 4 y -20 = 0 \)
\( 3y^2 -6y +10y -20 = 0 \)
\( (3y + 10) ( y - 2) \)
So, y = 2 or -3/10 but y = \( (3^x) \) cannot be negative; So,\( (3^x) \) = 2

x must be greater than 1/2 ( sqrt 3) = 1.732 but less than 1 (3^1 = 3)

A. 1/6 - not greater than 1/2
B. 1/5 - not greater than 1/2
C. 1/3 - not greater than 1/2
D. 2/3 - best choice
E. 9/10- too close to 1

IMO D is correct
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:27
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Kudos
Consider 3^x = a

Therefore, the equation can be written as:

3*a^2 + 4*a - 20 = 0

3*a^2 + 10*a - 6*a - 20 = 0

a*(3*a + 10) - 2*(3*a+10) = 0

(3*a + 10)*(a - 2) = 0

a = 2 or -10/3

Since 3^x can never be negative, -10/3 cannot be a solution

Therefore, 3^x = 2

Substituting the options:

A. 1/6

3^(1/6) = 2
3 = 2^6 --> Incorrect

B. 1/5

3^(1/5) = 2

3 = 2^5 --> Incorrect

C. 1/3

3^(1/3) = 2
3 = 2^3 --> Incorrect

D. 2/3

3^(2/3) = 2
3^2 = 2^3
8 = 9 --> values are aprroximately the same. Correct

E. 9/10

3^(9/10) = 2
3^9 = 2^10 --> Incorrect
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink] New post  24 Aug 2021, 08:33
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(D) is the answer IMO

Say \(3^x=y\)

\(3*y^2 + 4y -20 = 0 \)

\((y-2)(3y+10)=0 \)
y=2 or y=\( \frac{-10}{3}\)

y can not be negative

\(3^x\) = 2

We know \((3)^1/3\) = 1.4 so \((3)^2/3 \) must be closer to 2

(D) is the answer


[quote="Bunuel"]If \(3^{(2x+1)} + 4(3^x) – 20 = 0\), then what is approximate value of x ?

A. 1/6
B. 1/5
C. 1/3
D. 2/3
E. 9/10
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Re: GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) – 20 = 0, then what is [#permalink]
24 Aug 2021, 08:33
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