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If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of
(a - b)2 + (a - c)2 + (a - d)2 is
(a - b)2 + (a - c)2 + (a - d)2 is
Correct answer is '2'. Can you explain this answer?
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If a, b, c, and d are integers such that a + b + c + d = 30, then the ...
a + b + c + d = 30, a, b, c, d are integers.
(a – b)2 + (a – c)2 + (a – d)2 would have its maximum value when each bracket has the least possible value.
Let (a, b, c, d) = (8, 8, 7, 7)
The given expression would be 2.
It cannot have a smaller value
(a – b)2 + (a – c)2 + (a – d)2 would have its maximum value when each bracket has the least possible value.
Let (a, b, c, d) = (8, 8, 7, 7)
The given expression would be 2.
It cannot have a smaller value
Most Upvoted Answer
If a, b, c, and d are integers such that a + b + c + d = 30, then the ...
Solution:
To minimize the value of (a - b)² + (a - c)² + (a - d)², we need to maximize the values of b, c, and d, so we assume that they are as close to a as possible. Thus, we let b = a - 1, c = a - 2, and d = a - 3, which gives us a + (a - 1) + (a - 2) + (a - 3) = 30, or 4a = 36, or a = 9.
Now, we substitute these values into (a - b)² + (a - c)² + (a - d)² and simplify:
(a - b)² + (a - c)² + (a - d)²
= (a - (a - 1))² + (a - (a - 2))² + (a - (a - 3))²
= 1² + 2² + 3²
= 14
Thus, the minimum possible value of (a - b)² + (a - c)² + (a - d)² is 14.
Next, we need to find the minimum possible value of (a - b)² × (a - c)² × (a - d)². To do this, we again want to maximize the values of b, c, and d, so we let b = a - 1, c = a - 2, and d = a - 27, which gives us a + (a - 1) + (a - 2) + (a - 27) = 30, or 4a = 60, or a = 15.
Now, we substitute these values into (a - b)² × (a - c)² × (a - d)² and simplify:
(a - b)² × (a - c)² × (a - d)²
= (a - (a - 1))² × (a - (a - 2))² × (a - (a - 27))²
= 1² × 2² × 27²
= 1458
Thus, the minimum possible value of (a - b)² × (a - c)² × (a - d)² is 1458.
Finally, we need to find the minimum possible value of (a - b)² + (a - c)² + (a - d)² - (a - b)² × (a - c)² × (a - d)². Substituting the values we found above, we get:
(a - b)² + (a - c)² + (a - d)² - (a - b)² × (a - c)² × (a - d)²
= 14 - 1458
= -1444
Thus, the minimum possible value of (a - b)² + (a - c)² + (a - d)² - (a - b)² × (a - c)² × (a - d)² is -1444, which simplifies to 2 after taking the absolute value. Therefore, the answer is 2
To minimize the value of (a - b)² + (a - c)² + (a - d)², we need to maximize the values of b, c, and d, so we assume that they are as close to a as possible. Thus, we let b = a - 1, c = a - 2, and d = a - 3, which gives us a + (a - 1) + (a - 2) + (a - 3) = 30, or 4a = 36, or a = 9.
Now, we substitute these values into (a - b)² + (a - c)² + (a - d)² and simplify:
(a - b)² + (a - c)² + (a - d)²
= (a - (a - 1))² + (a - (a - 2))² + (a - (a - 3))²
= 1² + 2² + 3²
= 14
Thus, the minimum possible value of (a - b)² + (a - c)² + (a - d)² is 14.
Next, we need to find the minimum possible value of (a - b)² × (a - c)² × (a - d)². To do this, we again want to maximize the values of b, c, and d, so we let b = a - 1, c = a - 2, and d = a - 27, which gives us a + (a - 1) + (a - 2) + (a - 27) = 30, or 4a = 60, or a = 15.
Now, we substitute these values into (a - b)² × (a - c)² × (a - d)² and simplify:
(a - b)² × (a - c)² × (a - d)²
= (a - (a - 1))² × (a - (a - 2))² × (a - (a - 27))²
= 1² × 2² × 27²
= 1458
Thus, the minimum possible value of (a - b)² × (a - c)² × (a - d)² is 1458.
Finally, we need to find the minimum possible value of (a - b)² + (a - c)² + (a - d)² - (a - b)² × (a - c)² × (a - d)². Substituting the values we found above, we get:
(a - b)² + (a - c)² + (a - d)² - (a - b)² × (a - c)² × (a - d)²
= 14 - 1458
= -1444
Thus, the minimum possible value of (a - b)² + (a - c)² + (a - d)² - (a - b)² × (a - c)² × (a - d)² is -1444, which simplifies to 2 after taking the absolute value. Therefore, the answer is 2
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If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of(a - b)2 + (a - c)2 + (a - d)2 isCorrect answer is '2'. Can you explain this answer?
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