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11 Jun 2018, 06:12
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B
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Difficulty:
45%
(medium)
Question Stats:
75% (01:55) correct
25%
(01:37)
wrong
based on 40
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When \(a≠0\), how many solutions does the equation \(a(x+b)^2+c=0\) have?
1) \(bc=0\)
2) \(|b|+|c|=0\)
1) \(bc=0\)
2) \(|b|+|c|=0\)
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Expert reply
11 Jun 2018, 06:12
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Official Solution:
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.
There are three cases to consider:
Case 1: \(a > 0\), \(c > 0\) or \(a < 0\), \(c < 0\)
The equation has no roots.
Case 2: \(c = 0\).
The equation has only one root.
Case 3: \(a > 0\), \(c < 0\) or \(a < 0\), \(c > 0\)
The equation has two roots.
Condition 1):
If \(bc = 0\), then when
\(a = 1\), \(b = 0\), \(c = -1\), the equation has two roots, and when
\(a = 1\), \(b = 0\), \(c = 0\), the equation has one root.
As the question does not have a unique answer, condition 1) is not sufficient.
Condition 2)
\(|b| + |c| = 0\) ⇔ \(b = c = 0\).
Since \(c = 0\), the equation has only one root.
Condition 2) is sufficient.
Therefore, the answer is B.
Answer: B
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.
There are three cases to consider:
Case 1: \(a > 0\), \(c > 0\) or \(a < 0\), \(c < 0\)
The equation has no roots.
Case 2: \(c = 0\).
The equation has only one root.
Case 3: \(a > 0\), \(c < 0\) or \(a < 0\), \(c > 0\)
The equation has two roots.
Condition 1):
If \(bc = 0\), then when
\(a = 1\), \(b = 0\), \(c = -1\), the equation has two roots, and when
\(a = 1\), \(b = 0\), \(c = 0\), the equation has one root.
As the question does not have a unique answer, condition 1) is not sufficient.
Condition 2)
\(|b| + |c| = 0\) ⇔ \(b = c = 0\).
Since \(c = 0\), the equation has only one root.
Condition 2) is sufficient.
Therefore, the answer is B.
Answer: B
27 Jul 2018, 07:29
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Not sure if I get the break down of the question. Can you please explain how you broke it down in a different way?
