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P = the product of all x-values that satisfy the

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BrentGMATPrepNow
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P = the product of all x-values that satisfy the [#permalink] New post   Updated on: 09 Jan 2017, 13:32
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P = the product of all x-values that satisfy the equation (x²)^(x² - 2x + 1) = x^(3x² + x + 8)
Which of the following is true?

A) P < -14
B) -14 ≤ P < -4
C) -4 ≤ P < 4
D) 4 ≤ P < 14
E) 14 ≤ P

*kudos for all correct solutions
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C

Originally posted by BrentGMATPrepNow on 09 Jan 2017, 13:01.
Last edited by BrentGMATPrepNow on 09 Jan 2017, 13:32, edited 1 time in total.
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Re: P = the product of all x-values that satisfy the [#permalink] New post   09 Jan 2017, 13:27
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With complicated equations, especially those that include ugly exponents, it can be helpful to consider easy cases first, often 0 and 1.

Here, 1 satisfies the equation (zero doesn't since we end up with zero raised to a negative, which is the same as dividing by zero). So our answer needs to include 1.

C is the only option that includes 1, so that should be our answer.
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Re: P = the product of all x-values that satisfy the [#permalink] New post   10 Jan 2017, 11:33
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GMATPrepNow wrote:
P = the product of all x-values that satisfy the equation (x²)^(x² - 2x + 1) = x^(3x² + x + 8)
Which of the following is true?

A) P < -14
B) -14 ≤ P < -4
C) -4 ≤ P < 4
D) 4 ≤ P < 14
E) 14 ≤ P

*kudos for all correct solutions


IMPORTANT: If b^x = b^y, then x = y, as long as b ≠ 0, b ≠ 1 and b ≠ -1
For example, if we have 1^x = 1^y, we cannot conclude that x = y, since 1^x equals 1^y FOR ALL values of x and y.
So, although 1² = 1³, we can't then conclude that 2 = 3.


So, let's first see what happens when the base (x) equals 0.

If x = 0, then we get: (0²)^(0² - 2(0) + 1) = 0^(3(0²) + 0 + 8)
Simplify: 0^1 = 0^8
Evaluate: 0 = 0
Perfect! We know that x = 0 is one solution to the equation.
This means the PRODUCT of all of the solutions will be ZERO, regardless of the other solutions.
In other words, P = 0

So, the correct answer is C

Cheers,
Brent
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Re: P = the product of all x-values that satisfy the [#permalink] New post   09 Jan 2017, 13:34
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GMATAcademy wrote:
With complicated equations, especially those that include ugly exponents, it can be helpful to consider easy cases first, often 0 and 1.

Here, 1 satisfies the equation (zero doesn't since we end up with zero raised to a negative, which is the same as dividing by zero). So our answer needs to include 1.

C is the only option that includes 1, so that should be our answer.

Argh! You're totally right about zero raised to a negative.
I have edited the question to reflect the intent of my question (the correct answer is still C though!)
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Re: P = the product of all x-values that satisfy the [#permalink] New post   12 Jan 2017, 10:10
I simplified equation, results two x values
\((x^{2}) ^{x^{2} - 2x + 1}\) = \(x^{3x^{2} + x + 8}\)
On simplifying
\(0 = x^{2} + 5x + 6\)
x = -2 ; x = -3
??
Could anyone pls explain.Thanks
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Re: P = the product of all x-values that satisfy the [#permalink] New post   12 Jan 2017, 12:25
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kanusha wrote:
I simplified equation, results two x values
\((x^{2}) ^{x^{2} - 2x + 1}\) = \(x^{3x^{2} + x + 8}\)
On simplifying
\(0 = x^{2} + 5x + 6\)
x = -2 ; x = -3
??
Could anyone pls explain.Thanks


That's great work, but you need to keep going.
We also need to test whether the equation also holds true when the base (x) equals 0, 1 and -1
See my earlier post for this.

Once we've checked those values as well, we see that all possible solutions are x = -2, -3, 0, 1 and -1
So, P = (-2)(-3)(0)(1)(-1) = 0
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Re: P = the product of all x-values that satisfy the [#permalink] New post   24 Dec 2018, 21:10
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GMATPrepNow wrote:
GMATPrepNow wrote:
P = the product of all x-values that satisfy the equation (x²)^(x² - 2x + 1) = x^(3x² + x + 8)
Which of the following is true?

A) P < -14
B) -14 ≤ P < -4
C) -4 ≤ P < 4
D) 4 ≤ P < 14
E) 14 ≤ P

*kudos for all correct solutions


IMPORTANT: If b^x = b^y, then x = y, as long as b ≠ 0, b ≠ 1 and b ≠ -1
For example, if we have 1^x = 1^y, we cannot conclude that x = y, since 1^x equals 1^y FOR ALL values of x and y.
So, although 1² = 1³, we can't then conclude that 2 = 3.


So, let's first see what happens when the base (x) equals 0.

If x = 0, then we get: (0²)^(0² - 2(0) + 1) = 0^(3(0²) + 0 + 8)
Simplify: 0^1 = 0^8
Evaluate: 0 = 0
Perfect! We know that x = 0 is one solution to the equation.
This means the PRODUCT of all of the solutions will be ZERO, regardless of the other solutions.
In other words, P = 0

So, the correct answer is C

Cheers,
Brent



But that give both C&E as answer

Posted from my mobile device
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Re: P = the product of all x-values that satisfy the [#permalink] New post   25 Dec 2018, 03:26
Hi,

If I simplify the equations above, I will arrive at: X^2+5x+6 = 0

The product of the solutions can be determined by the rule that the product equals = c/a when the equation is in the form ax^2+bx+c=0. Using this will lead us to the product being 6.

Can anyone from the experts tell what went wrong here please?
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Re: P = the product of all x-values that satisfy the [#permalink] New post   30 Dec 2018, 10:35
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UdayPratapSingh wrote:
GMATPrepNow wrote:
GMATPrepNow wrote:
P = the product of all x-values that satisfy the equation (x²)^(x² - 2x + 1) = x^(3x² + x + 8)
Which of the following is true?

A) P < -14
B) -14 ≤ P < -4
C) -4 ≤ P < 4
D) 4 ≤ P < 14
E) 14 ≤ P

*kudos for all correct solutions


IMPORTANT: If b^x = b^y, then x = y, as long as b ≠ 0, b ≠ 1 and b ≠ -1
For example, if we have 1^x = 1^y, we cannot conclude that x = y, since 1^x equals 1^y FOR ALL values of x and y.
So, although 1² = 1³, we can't then conclude that 2 = 3.


So, let's first see what happens when the base (x) equals 0.

If x = 0, then we get: (0²)^(0² - 2(0) + 1) = 0^(3(0²) + 0 + 8)
Simplify: 0^1 = 0^8
Evaluate: 0 = 0
Perfect! We know that x = 0 is one solution to the equation.
This means the PRODUCT of all of the solutions will be ZERO, regardless of the other solutions.
In other words, P = 0

So, the correct answer is C

Cheers,
Brent



But that give both C&E as answer

Posted from my mobile device


If P = 0, answer choice E doesn't work, since it says P is GREATER THAN or equal to 14

Cheers,
Brent
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Re: P = the product of all x-values that satisfy the [#permalink] New post   21 Oct 2019, 06:29
GMATPrepNow wrote:
P = the product of all x-values that satisfy the equation (x²)^(x² - 2x + 1) = x^(3x² + x + 8)
Which of the following is true?

A) P < -14
B) -14 ≤ P < -4
C) -4 ≤ P < 4
D) 4 ≤ P < 14
E) 14 ≤ P


\(x^{2(x^2-2x+1)}=x^{(3x^2+x+8)}\)

\(base≠(0,1,-1):{2(x^2-2x+1)}={(3x^2+x+8)}…x^2+5x+6=0…(x+3)(x+2)=0\)
\(base≠(0,1,-1): (x+3)(x+2)=0…x=(-3,-2)\)

\(base=(0,1,-1):x^{2(x^2-2x+1)}=x^{(3x^2+x+8)}…(0,1,-1)^{anything}=(0,1,-1)^{anything}\)
\(base=(0,1,-1):x=(0,1,-1,anything)\)

\(P=product(-3,-2,0,1,-1,anything)=(0)(-3,-2,1,-1,anything)=0\)

Answer (C)
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Re: P = the product of all x-values that satisfy the [#permalink] New post   12 Dec 2023, 19:22
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