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# When a≠0, how many solutions does the equation

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6028
GMAT 1: 760 Q51 V42
GPA: 3.82
When a≠0, how many solutions does the equation  [#permalink]

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23 Jan 2018, 01:10
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Difficulty:

55% (hard)

Question Stats:

61% (01:05) correct 39% (01:34) wrong based on 88 sessions

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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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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" PS Forum Moderator Joined: 25 Feb 2013 Posts: 1201 Location: India GPA: 3.82 When a≠0, how many solutions does the equation [#permalink] ### Show Tags 23 Jan 2018, 02:41 1 1 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 Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6028 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: When a≠0, how many solutions does the equation [#permalink] ### Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: When a≠0, how many solutions does the equation &nbs [#permalink] 25 Jan 2018, 04:03
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