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12 May 2024, 03:09
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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
12 May 2024, 03:09
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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:
(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
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
06 Nov 2024, 03:28
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If I consider all the values to be positive and open the equation, then it will come like this
a+b+c-2x+x^2 -a-b-c=-c
x^2-2x=-c
Now, if I put x value as 2 then I get C value as 0 So, the answer of the question is 2.
However, if I put x value as 3, then value of c=-3 then I am getting the answer as 0.
Hence, I marked the ans choice as E.
Can you please guide here?
Thanks
a+b+c-2x+x^2 -a-b-c=-c
x^2-2x=-c
Now, if I put x value as 2 then I get C value as 0 So, the answer of the question is 2.
However, if I put x value as 3, then value of c=-3 then I am getting the answer as 0.
Hence, I marked the ans choice as E.
Can you please guide here?
Thanks
