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When a≠0, how many solutions does the equation

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New post 23 Jan 2018, 01:10
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[GMAT math practice question]

When \(a≠0\), how many solutions does the equation \(a(x+b)^2+c=0\) have?

\(1) bc=0\)
\(2) |b|+|c|=0\)

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New post 23 Jan 2018, 02:41
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MathRevolution wrote:
[GMAT math practice question]

When \(a≠0\), how many solutions does the equation \(a(x+b)^2+c=0\) have?

\(1) bc=0\)
\(2) |b|+|c|=0\)


Statement 1: implies either \(b=0\) or \(c=0\)

if \(c=0\), then we have \(x=-b\)

but if \(b=0\), then we have \(ax^2=-c\), there could be multiple solutions. Insufficient

Statement 2: sum of two positive numbers is \(0\) only when both themselves are \(0\)

\(=> b=c=0\)

hence our equation becomes \(ax^2=0\), this is only possible when \(x=0\) because \(a≠0\). So we have \(1\) solution here. Sufficient

Option B
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Re: When a≠0, how many solutions does the equation  [#permalink]

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New post 25 Jan 2018, 04:03
=>

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
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Re: When a≠0, how many solutions does the equation  [#permalink]

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New post 23 Aug 2019, 00:55
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Re: When a≠0, how many solutions does the equation   [#permalink] 23 Aug 2019, 00:55
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