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If a*b 0, and (x a)(x + b) = 0, is b = a? (1) x + a = 0 (2) x+b=0

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Bunuel
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If a*b 0, and (x a)(x + b) = 0, is b = a? (1) x + a = 0 (2) x+b=0 [#permalink] New post   10 Feb 2023, 02:30
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53% (01:52) correct 47% (02:19) wrong based on 58 sessions
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If \(a*b ≠ 0\), and \((x − a)(x + b) = 0\), is \(b = a\) ?

(1) \(x + a = 0\)

(2) \(x + b = 0\)



700-Level Question!!!


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A
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If a*b 0, and (x a)(x + b) = 0, is b = a? (1) x + a = 0 (2) x+b=0 [#permalink] New post   10 Feb 2023, 05:54
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If a∗b≠0 and (x−a)(x+b)=0, is b=a ?

a*b ≠ 0 implies that both a and b are not equal to 0.

(1) x+a = 0
=> x = -a
substituting in given equation, we get
(-a-a)(-a+b)=0
=> -2a(b-a)=0
but we have deduced above that a ≠ 0 so (b-a) must be equal to 0
therefore a =b. Statement 1 is sufficient

(2) x+b=0
=>x = -b
substituting in given equation, we get
(-b-a)(-b+b) = 0
-(b+a)(0) = 0
therefore we cannot deduce anything further.
Another way to approach is - its already given that (x-a)(x+b) = 0
which implies either (x-a) = 0 or x +b =0 or both (x-a) and (x+b) =0
so statement 2 is not giving any extra information from what we already know.
therefore, statement 2 is Not Sufficient
Answer A
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Re: If a*b 0, and (x a)(x + b) = 0, is b = a? (1) x + a = 0 (2) x+b=0 [#permalink] New post   14 Mar 2023, 01:57
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Bunuel wrote:
If \(a*b ≠ 0\), and \((x − a)(x + b) = 0\), is \(b = a\) ?
(1) \(x + a = 0\)
(2) \(x + b = 0\)

Solution:
Pre Analysis:
  • \(a*b ≠ 0\) means neither a or b is equal to 0
  • \((x − a)(x + b) = 0\) means the value of x is either a or -b
  • We are asked if \(a=b\) or not

Statement 1: \(x + a = 0\)
  • From this, we can say \(x=-a\) and \(x≠a\)
  • From pre-analysis we know if \(x≠a\), then \(x=-b\)
  • So, we know \(x=-a=-b\) or \(a=b\)
  • Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: \(x + b = 0\)
  • From this, we can say \(x=-b\)
  • No information on a

Hence the right answer is Option A
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Re: If a*b 0, and (x a)(x + b) = 0, is b = a? (1) x + a = 0 (2) x+b=0 [#permalink] New post   14 Mar 2023, 02:08
Made a silly mistake, but obviously liked the sum!

Statement 1 : If x+a=0 then obviously (x-a) is not equal to 0. Out of (x-a) and (x+b), one needs to be 0 and therefore (x+b) is 0. So, x=-b and x=-a. Therefore b must be equal to a.

Statement 2 : If x+b=0, then x=-b. But we have no information about 'a'.

Therefore, statement 1 is sufficient.
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Re: If a*b 0, and (x a)(x + b) = 0, is b = a? (1) x + a = 0 (2) x+b=0 [#permalink]
14 Mar 2023, 02:08
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