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If \(x\) is a prime number and \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|\), what is the value of \(x + c\)?
A. 0
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
A. 0
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
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01 Jun 2024, 12:30
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Official Solution:
If \(x\) is a prime number and \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|\), what is the value of \(x + c\)?
A. 0
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
Re-write \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|\) as: \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} + |c| = 0\). We have that the sum of three non-negative values (the absolute value of some number, the square root of a square, and the absolute value of a number) is 0. For this to be true, each of the three must be 0.
So:
Equate (i) and (ii): \(2x = x^2\). This yields \(x = 0\) or \(x = 2\). It's given that \(x\) is a prime number, so \(x = 2\).
Answer: C
If \(x\) is a prime number and \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|\), what is the value of \(x + c\)?
A. 0
B. 1
C. 2
D. 3
E. Cannot be determined from the given information
Re-write \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|\) as: \(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} + |c| = 0\). We have that the sum of three non-negative values (the absolute value of some number, the square root of a square, and the absolute value of a number) is 0. For this to be true, each of the three must be 0.
So:
- (i) \(a + b + c - 2x=0\), which gives \(a + b + c=2x\);
(ii) \(x^2 - a - b - c=0\), which gives \(a + b + c=x^2\);
(iii) \(|c|=0\), which gives \(c=0\).
Equate (i) and (ii): \(2x = x^2\). This yields \(x = 0\) or \(x = 2\). It's given that \(x\) is a prime number, so \(x = 2\).
- \(x + c=2+0=2\).
Answer: C
General Discussion
14 May 2024, 01:57
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Bunuel
\(|a + b + c - 2x| + \sqrt{(x^2 - a - b - c)^2} = - |c|\)
Rule : \(\sqrt{(m)^2} = |m|\)
\(|a + b + c - 2x| + |(x^2 - a - b - c)| = - |c|\)
The left hand side of the equation must be a non-negative number as we are adding two values which are non-negative.
Hence, the value of \(c\) cannot be positive and must be zero.
\(c = 0\)
\(|a + b + c - 2x| + |(x^2 - a - b - c)| = 0\)
As the sum equals zero, each of the term must be equal to zero
\(a + b + c - 2x = 0\) and \(x^2 - a - b - c = 0\)
\(a + b + c - 2x = 0\)
\(a + b + c = 2x \) → (1)
\(x^2 - a - b - c = 0\)
\(x^2 = a + b + c \) → (2)
(1) = (2)
\(x^2 = 2x\)
As \(x\) is a prime number, we know that \(x \neq 0\), hence dividing both sides by \(x\)
\(x = 2\)
\(x + c = 2 + 0 = 2\)
Option C
