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23 Apr 2024, 01:30
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83% (02:42) correct
17%
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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
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04 May 2024, 22:45
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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
General Discussion
23 Apr 2024, 03:22
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\(|x^2 - 2| - |x - 2| = -2\)
Let's obtain the critical points of the equation.
Let's plot these over a number line
--- Region 1 --- \(-\sqrt{2}\) --- Region 2 --- \(\sqrt{2}\) --- Region 3 --- 2 --- Region 4 ---
Region 1 (\(x < -\sqrt{2}\))
\(|x^2 - 2| - |x - 2| = -2\)
\(x^2 - 2 + x - 2 = -2\)
\(x^2 - 2 + x = 0\)
Value of x = -2 (we ignore x = 1, as the value falls out of range.)
Region 2 (\(-\sqrt{2} \leq x \leq \sqrt{2}\))
\(|x^2 - 2| - |x - 2| = -2\)
\(-x^2 + 2 + x - 2 = -2 \)
\(-x^2 + x + -2 = 0\)
Between 1 and -1, -1 satisfies the equation.
Region 3 (\(\sqrt{2} < x < 2\))
We can ignore this region, as there are no integers in this region.
Region 4 (\(x \geq 2\))
\(|x^2 - 2| - |x - 2| = -2\)
\(x^2 - 2 - x + 2 = -2\)
\(x^2 - x + 2 = 0\)
No integer value of \(x\) satisfies the equation. Hence, we can ignore this region.
Hence, the only possible value of x = {-1, -2}
For both values of x, \(x^4 - 5x^2 + 4 = 0\)
Option C
Let's obtain the critical points of the equation.
- \(x^2 - 2 = 0\)
- ⇒ \(x = \pm \sqrt{2}\)
- \(x - 2 = 0\)
- ⇒ \(x = 2\)
Let's plot these over a number line
--- Region 1 --- \(-\sqrt{2}\) --- Region 2 --- \(\sqrt{2}\) --- Region 3 --- 2 --- Region 4 ---
Region 1 (\(x < -\sqrt{2}\))
\(|x^2 - 2| - |x - 2| = -2\)
\(x^2 - 2 + x - 2 = -2\)
\(x^2 - 2 + x = 0\)
Value of x = -2 (we ignore x = 1, as the value falls out of range.)
Region 2 (\(-\sqrt{2} \leq x \leq \sqrt{2}\))
\(|x^2 - 2| - |x - 2| = -2\)
\(-x^2 + 2 + x - 2 = -2 \)
\(-x^2 + x + -2 = 0\)
Between 1 and -1, -1 satisfies the equation.
Region 3 (\(\sqrt{2} < x < 2\))
We can ignore this region, as there are no integers in this region.
Region 4 (\(x \geq 2\))
\(|x^2 - 2| - |x - 2| = -2\)
\(x^2 - 2 - x + 2 = -2\)
\(x^2 - x + 2 = 0\)
No integer value of \(x\) satisfies the equation. Hence, we can ignore this region.
Hence, the only possible value of x = {-1, -2}
For both values of x, \(x^4 - 5x^2 + 4 = 0\)
Option C
