Simplifying Error - (2)
a + b − 2 a b
Which of the following options equals the above expression if a and b are real numbers?
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2 solutions
It's a bit nonsensical to say that "the answer cannot be determined"... of course it can.
I understand that you don't want to get into a discussion of "principal values" of roots, but if you had offered " a − b OR b − a " as a possible answer... that would have been correct [writing the expression as you did initially, as a + b − 2 a b ]
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I still think that answer should be "Cannot be determined", as we cannot determine it without knowing the relation between a and b. "None of these", on the other hand implies that the option of cannot be determined is false , which implies that it can be determined , which is absolutely false. Also, I think the domain of a and b should have been mentioned in the question.
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But it can be determined..... it is the principal value of these two roots, just like 4 = 2 rather than -2... would you say that 4 cannot be determined"?
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@Otto Bretscher – But Sir in this question, without the knowledge of a and b, how can we determine it? (I mean, how we can reach the principal value in this case?)
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@Abhijeet Verma – Just as 4 is defined as the positive root of 4 , so, more generally, w for a complex number w is defines as the principal root, that is, the root with its argument on the interval ( − π / 2 , π / 2 ] . Think about it!
I totally agree with you
Thanks, I've added a choice "None of these choices" and has changed it to the correct answer.
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great...thanks
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@Otto Bretscher – I find it amusing that now, in the pop-up box at the lower right of the question page, it states that "101% of people couldn't solve this problem". Perhaps this should warrant a level 6 rating, rather than a level 2. :)
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@Brian Charlesworth – LOL Sir , this problem has :
123 views (100%) 93 attempts (76%) 0 solvers (0%)
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@Nihar Mahajan – I wonder how it got the 101% then? :) And why did someone downvote your comment?
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@Brian Charlesworth – See the downvoter got scared ... :P
Sir, when I had answered it there was no option saying "none of these choices" so I clicked "cannot be detrrmined", now what about that?
Please don't take my comment agresively ⌣ ¨
Actually I was confused whether to have option of "the answer cannot be determined" or "none of the choices".
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But, it doesn't matter whether a, b > or < 0, since we already know that a and b themselves are real. We are not actually squaring surds over here... We are just rooting perfect squares, which BTW, can be rooted irrespective of the fact that whether they are rational or irrational. We can easily say that the above expression equals sqrt((sqrt a - sqrt b)^2), which equals ±(sqrt a - sqrt b)… It also does not depend on whether a > b or b > a, since there's nothing wrong with the possibility of -ve answers. And Nihar Mahajan, if this problem has no solvers, you really ought to consider the fact that you've framed it wrong. xD
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@Debneil Nag Chowdhury – Square root is always positive. x 2 = ∣ x ∣
( a − b ) 2 = ± ( a − b )
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@Abhijeet Verma – Yeah, but that thing under the main root can be both (√a - √b)² and (√b - √a)². So when we root these two, we will get both √a - √b and √b - √a as correct answers. Concisely, that's nothing but ±(√a - √b).
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@Debneil Nag Chowdhury – Actually, you are again mistaken. When the square term comes out of the root, only the positive value is correct. For example , 4 = + 2 4 = − 2
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@Abhijeet Verma – Thats not what I meant... a + b -2√ab =(√a)² + (√b)² - 2.√a.√b =(√a - √b)² = (√b - √a)²
In other words, Both of the above squares equal a + b - 2√ab, as (x-y)² = x² + y² - 2xy = (y-x)².
And, as for your example, 2×2 = -2×-2 = 4, which implies √4 = ±2, as a quadratic expression always has two solutions, the expression here being ax² + bx + c = 4, where x=2, a=1, and b=c=0
So you are actually yourself confused...
Good question and great explanation!
Bonus Question: What if they are imaginary numbers? i.e. a = b = i
so when you look at it., isnt +or - ( \sqrt{a} - \sqrt{b} ) the real answer.? because its either \sqrt{a} - \sqrt{b} or \sqrt{b} - \sqrt{a} .
this is not a question. It's a trick. it's just like saying: "There are 10 kinds of people. The ones that laughs with this joke; the ones that did not laugh; and the people who expected a binary joke." Ok, if you let me complete my answer, it would be a − b in case that a , b ≥ 0 , which is not one of the possible answers. Too bad, I've answered 2 questions like this lately, and I feel disappointed. My interest in solving problems is big. My desire of being cheated by little tricky nuances (and very neccessary, perphaps) is very, very small. If I continue finding annoying questions, my interest for this page will tend to zero. Or, do I have to add, to be very precise, once we model my interest by a real positive number, it will tend to zero??
But the last option can be a solution.If a > b then the value becomes sqrt(a) - sqrt(b) and if b>a , then the value becomes sqrt(b) - sqrt(a).
The solution is : |(a^(1/2))-(b^(1/2))|
We do not know whether a , b > 0 o r a , b < 0 . We also don't know whether b > a . Thus , the answer is none of the choices.