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equation with fourth root

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GreenlightTestPrep
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equation with fourth root [#permalink] New post   Updated on: 31 May 2019, 13:03
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\(x = \sqrt[4]{(x^3 + 6x^2)}\)

Quantity A
Quantity B
sum of all possible values of x
1


A) The quantity in Column A is greater.
B) The quantity in Column B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
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A

Originally posted by GreenlightTestPrep on 02 Feb 2019, 07:07.
Last edited by GreenlightTestPrep on 31 May 2019, 13:03, edited 2 times in total.
Edited by Carcass
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GreenlightTestPrep
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Re: equation with fourth root [#permalink] New post   03 Dec 2021, 06:19
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JKGrowling wrote:
This is incorrect the answer should be C.
(16)^1/2 has 2 solutions, 4 and -4
For a better representation, check the graphical representation of y = x^2. It is a parabola, with 2 x-intercepts(roots) centered around the y axis. The graph for y = x^4 is similar, except that is steeper because y rises faster than the run of x since it is a higher power.


Be careful, although there are two x-values (2 and -2) that satisfy the equation x^4 = 16, the fourth root notation tells us to take the positive root only.
So, ∜(16) = 2 (not -2)

If you want a graphical solution, it's better to graph the equation y = ∜x
If you do, you'll get something like this:
Image

As you can see, there's only one value of y corresponding to x = 16
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GreenlightTestPrep
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Re: equation with fourth root [#permalink] New post   02 Feb 2019, 07:09
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GreenlightTestPrep wrote:
x = ∜(x³ + 6x²)

Quantity A
Quantity B
sum of all possible values of x
1


A) The quantity in Column A is greater.
B) The quantity in Column B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.


Given: x = ∜(x³ + 6x²)
Raise both sides to the power of 4 to get: x⁴ = x³ + 6x²
Rewrite equation as follows: x⁴ - x³ - 6x² = 0
Factor to get: x²(x² - x - 6) = 0
Factor more to get: x²(x - 3)(x + 2) = 0

This means x² = 0, (x - 3) = 0, or (x + 2) = 0
So, the possible solutions are x = 0, x = 3 and x = -2

IMPORTANT: At this point, we should plug each answer choice back into the equation to identify any possible extraneous roots.
Test x = 0. We get: 0 = ∜(0³ + 6*0²) = ∜(0). WORKS
Test x = 3. We get: 3 = ∜(3³ + 6*3²) = ∜(81). WORKS
Test x = -2. We get: -2 = ∜(-2³ + 6*-2²) = ∜(16). DOESN’T work.

So, the only VALID solutions are x = 0 and x = 3
So, the SUM of all possible values of x = 0 + 3 = 3

We get:
QUANTITY A: 3
QUANTITY B: 1

Answer: A

Cheers,
Brent
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Re: equation with fourth root [#permalink] New post   03 Mar 2019, 08:25
Test x = -2. We get: -2 = ∜(-2³ + 6*-2²) = ∜(16). DOESN’T work. Why? please explain .
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Re: equation with fourth root [#permalink] New post   04 Mar 2019, 09:20
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JelalHossain wrote:
Test x = -2. We get: -2 = ∜(-2³ + 6*-2²) = ∜(16). DOESN’T work. Why? please explain .



Square root of a number always gives the positive root.
For example,
x=√16 will be 4 and not -4.

To remember:

√x²=|x| (always)
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Re: equation with fourth root [#permalink] New post   04 Mar 2019, 09:41
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JelalHossain wrote:
Test x = -2. We get: -2 = ∜(-2³ + 6*-2²) = ∜(16). DOESN’T work. Why? please explain .


∜(any positive number) = some POSITIVE number

So, ∜(16) = 2 (not -2)
So, the equation -2 = ∜(16) does not hold true.

Does that help?

Cheers,
Brent
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Re: equation with fourth root [#permalink] New post   04 Mar 2019, 10:45
the answer is A right?
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Re: equation with fourth root [#permalink] New post   04 Mar 2019, 11:33
NoorJassem wrote:
the answer is A right?


Yes, that's correct.
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Re: equation with fourth root [#permalink] New post   05 Aug 2021, 07:52
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square root of a positive number can be positive and negative. Hence the answer is C
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Re: equation with fourth root [#permalink] New post   03 Dec 2021, 02:12
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This is incorrect the answer should be C.
(16)^1/2 has 2 solutions, 4 and -4
For a better representation, check the graphical representation of y = x^2. It is a parabola, with 2 x-intercepts(roots) centered around the y axis. The graph for y = x^4 is similar, except that is steeper because y rises faster than the run of x since it is a higher power.
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Re: equation with fourth root [#permalink] New post   03 Dec 2021, 07:43
How to know that whether it works or not?
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Re: equation with fourth root [#permalink] New post   21 Sep 2023, 09:40
Cant we consisder X^2 = t and then solve
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Re: equation with fourth root [#permalink] New post   21 Sep 2023, 10:13
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and once substituted then what ?
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