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# What is the sum of all solutions to the equation

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Joined: 12 Sep 2015
Posts: 4015
What is the sum of all solutions to the equation  [#permalink]

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28 Feb 2017, 09:31
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00:00

Difficulty:

65% (hard)

Question Stats:

65% (02:29) correct 35% (02:45) wrong based on 155 sessions

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What is the sum of all solutions to the equation $$3x^{2/3} = 54 + 9x^{1/3}$$

A) 63
B) 189
C) 216
D) 243
E) 567

*kudos for all correct solutions

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What is the sum of all solutions to the equation  [#permalink]

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01 Mar 2017, 06:46
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GMATPrepNow wrote:
What is the sum of all solutions to the equation $$3x^{2/3} = 54 + 9x^{1/3}$$

A) 63
B) 189
C) 216
D) 243
E) 567

We should start by recognizing that this is a QUADRATIC EQUATION in disguise.
Notice that $$x^{2/3} = (x^{1/3})^{2}$$
So let's let $$x^{1/3}$$ = k, and replace $$x^{1/3}$$ with k to get: 3k² = 54 + 9k
Rearrange to get: 3k² - 9k - 54 = 0
Factor: 3(k² - 3k - 18) = 0
Factor more: 3(k - 6)(k + 3) = 0
So, the solutions are k = 6 and k = -3

Since k = $$x^{1/3}$$, we can write: $$x^{1/3}$$ = 6 and $$x^{1/3}$$ = -3

If $$x^{1/3}$$ = 6, then x = 6³ = 216
If $$x^{1/3}$$ = -3, then x = (-3)³ = -27

So, the sum of all solutions = 216 + (-27) = 189

Cheers,
Brent
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##### General Discussion
Marshall & McDonough Moderator
Joined: 13 Apr 2015
Posts: 1684
Location: India
Re: What is the sum of all solutions to the equation  [#permalink]

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28 Feb 2017, 09:38
3
Let x^(1/3) = a

3a^2 = 54 + 9a
a^2 = 18 + 3a
a^2 - 3a - 18 = 0

Upon solving, we get a = 6 or -3
x^(1/3) = 6 --> x = 6^3 = 216
x^(1/3) = -3 --> x = -27

Sum of the solutions = 216 - 27 = 189

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Re: What is the sum of all solutions to the equation  [#permalink]

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16 Jul 2017, 13:52
1
What is the sum of all solutions to the equation $$3x^{2/3} = 54 + 9x^{1/3}$$

Let $$x^{1/3} = a$$

$$3a^2 = 54 + 9a$$

$$3a^2 - 9a - 54 = 0$$

$$a^2 - 3a - 18 = 0$$

$$(a - 6) (a + 3) = 0$$

$$a = 6$$ Or

$$a = -3$$

As, $$x^{1/3} = a$$

$$x^{1/3} = 6$$

$$x = 6 * 6 * 6 = 216$$

$$x^{1/3} = -3$$

$$x = -3 * -3 * -3 = -27$$

So, sum of the roots $$= 216 + (-27) = 216 - 27 = 189$$

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Joined: 27 Apr 2016
Posts: 6
Re: What is the sum of all solutions to the equation  [#permalink]

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26 Feb 2018, 20:05
GMATPrepNow wrote:
GMATPrepNow wrote:
What is the sum of all solutions to the equation $$3x^{2/3} = 54 + 9x^{1/3}$$

A) 63
B) 189
C) 216
D) 243
E) 567

We should start by recognizing that this is a QUADRATIC EQUATION in disguise.
Notice that $$x^{2/3} = (x^{1/3})^{2}$$
So let's let $$x^{1/3}$$ = k, and replace $$x^{1/3}$$ with k to get: 3k² = 54 + 9k
Rearrange to get: 3k² - 9k - 54 = 0
Factor: 3(k² - 3k - 18) = 0
Factor more: 3(k - 6)(k + 3) = 0
So, the solutions are k = 6 and k = -3

Since k = $$x^{1/3}$$, we can write: $$x^{1/3}$$ = 6 and $$x^{1/3}$$ = -3

If $$x^{1/3}$$ = 6, then x = 6³ = 216
If $$x^{1/3}$$ = -3, then x = (-3)³ = -27

So, the sum of all solutions = 216 + (-27) = 189

Cheers,
Brent

thanks for the explanation - I got to the part where k = -6 or 3. Why are we not plugging these numbers back into the equation $$3x^{2/3} = 54 + 9x^{1/3}$$ replacing x for -6 and 3?
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Re: What is the sum of all solutions to the equation  [#permalink]

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27 May 2019, 10:23
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Re: What is the sum of all solutions to the equation   [#permalink] 27 May 2019, 10:23
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