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22 Apr 2024, 07:39
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If \(x\) is a non-zero integer and \(|x^2 - 2| - |x - 2| = -2\), what is the value of \(x^4 - 5x^2 + 4\)?
A. -4
B. -2
C. 0
D. 2
E. 4
A. -4
B. -2
C. 0
D. 2
E. 4
22 Apr 2024, 07:39
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Official Solution:
If \(x\) is a non-zero integer and \(|x^2 - 2| - |x - 2| = -2\), what is the value of \(x^4 - 5x^2 + 4\)?
A. -4
B. -2
C. 0
D. 2
E. 4
Re-arrange \(|x^2 - 2| - |x - 2| = -2\) to \(|x^2 - 2| + 2 = |x - 2|\).
Notice that if \(x\) is a positive integer or 0, then \(|x^2 - 2|\) is more than or equal to \(|x - 2|\) (they are equal when \(x=0\) or \(x=1\)). Thus, \(|x^2 - 2| + 2\) will always be greater than \(|x - 2|\). Therefore, \(x\) cannot be positive.
Now, when \(x < 0\), \(|x - 2| = -(x-2) = 2 - x\). In this case, we'd have \(|x^2 - 2| - (2 - x) = -2\), which simplifies to \(|x^2 - 2| = -x\).
Square both sides: \(x^4 - 4x^2 + 4 = x^2\).
Re-arrange: \(x^4 - 5x^2 + 4 = 0\). This is exactly what we were asked to find.
Answer: C
If \(x\) is a non-zero integer and \(|x^2 - 2| - |x - 2| = -2\), what is the value of \(x^4 - 5x^2 + 4\)?
A. -4
B. -2
C. 0
D. 2
E. 4
Re-arrange \(|x^2 - 2| - |x - 2| = -2\) to \(|x^2 - 2| + 2 = |x - 2|\).
Notice that if \(x\) is a positive integer or 0, then \(|x^2 - 2|\) is more than or equal to \(|x - 2|\) (they are equal when \(x=0\) or \(x=1\)). Thus, \(|x^2 - 2| + 2\) will always be greater than \(|x - 2|\). Therefore, \(x\) cannot be positive.
Now, when \(x < 0\), \(|x - 2| = -(x-2) = 2 - x\). In this case, we'd have \(|x^2 - 2| - (2 - x) = -2\), which simplifies to \(|x^2 - 2| = -x\).
Square both sides: \(x^4 - 4x^2 + 4 = x^2\).
Re-arrange: \(x^4 - 5x^2 + 4 = 0\). This is exactly what we were asked to find.
Answer: C
20 Oct 2024, 11:01
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I used smart numbers. I started with x=1 and then x=-1. x=-1 is a solution and then replaced the numbers (-1)^4-5(-1)^2+4= 1 - 5 + 4 = 0
MBAToronto2024
