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28 Feb 2017, 09:31
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B
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Difficulty:
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Question Stats:
67% (02:29) correct
33%
(02:58)
wrong
based on 401
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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
A) 63
B) 189
C) 216
D) 243
E) 567
*kudos for all correct solutions
01 Mar 2017, 06:46
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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
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
Answer: B
Cheers,
Brent
28 Feb 2017, 09:38
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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
Answer: B
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
Answer: B
