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What's wrong in it ???????????????

Main post link -> http://pibyzero.blogspot.in/2013/05/q.html

Q. A natural number is greater than three times its square root by 4. Find the number.

 let, the number be x.

hence according to the condition- x = 3√x + 4

x-4 = 3√x

x² - 8x + 16 = 9x

x² - 17x + 16 = 0

(x-16)(x-1) = 0

x = 16 or x = 1

But when the negative value of square root of 16 and that positive of 1 is taken the equation is not satisfied. WHY ?????????????? Where is the mistake ?

Note by Rushikesh Jogdand
4 years, 6 months ago

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Squaring introduces an extraneous root. For example, if one plugs 1 into x-4=3sqrt(x), one gets -3=3, which is clearly not true, but the statement resulting from squaring both sides would be true.

Bob Krueger - 4 years, 6 months ago

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But if the same question is asked in examination what should be the answer ?

Rushikesh Jogdand - 4 years, 6 months ago

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I'm not going to help you cheat. You can figure that one out for yourself.

Bob Krueger - 4 years, 6 months ago

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@Bob Krueger How do you infer that he is trying to cheat? Please understand the argument he has proposed more clearly.

Aditya Parson - 4 years, 6 months ago

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@Aditya Parson agreed.........:)

Riya Gupta - 4 years, 6 months ago

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@Bob Krueger It's called a hypothetical, Bob. No need for the attitude.

Tim Ye - 4 years, 6 months ago

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@Tim Ye You never know any more. I've seen too many of these discussions say "URGENT" or other key words like that which give me the impression they need it for an exam or school.

Bob Krueger - 4 years, 6 months ago

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@Bob Krueger Hey You all are in misconception that it is a problem in any exam or homework. But it is not so . In fact it is a problem in the Question bank (whose problems are not forced to solve) in my algebra book[10 th std. algebra book of Maharashtra State Board of Secondary and Higher Education]. These problems are only for practice. But I had asked a suggestion from you on it because I was confused how to write the answer of same kind of confusing question if asked in real Exams. I hope you will concentrate about the question and not the asker's purpose. I was asking for the steps and reasons and not the solution

Rushikesh Jogdand - 4 years, 6 months ago

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Since it is not stated that the square root of the natural number should be positive or negative. I think \(1\) will also satisfy the equation. \(1=3\sqrt{1}+4\)

\(\Rightarrow 1=-3+4\)

\(1=1\)

which clearly satisfies the above conditions. So both 1 and 16 qualify as answers.

Aditya Parson - 4 years, 6 months ago

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Hey You all are in misconception that it is a problem in any exam or homework. But it is not so . In fact it is a problem in the Question bank (whose problems are not forced to solve) in my algebra book[10 th std. algebra book of Maharashtra State Board of Secondary and Higher Education]. These problems are only for practice. But I had asked a suggestion from you on it because I was confused how to write the answer of same kind of confusing question if asked in real Exams. I hope you will concentrate about the question and not the asker's purpose.

Rushikesh Jogdand - 4 years, 6 months ago

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when we square any equation with a variable....the no. of roots of...increases automatically.....in those some roots does not satisfies the equation.....

also if u check out....√(-4) = 4i

which is a complex no. and solutions can only be natural no....hence -4 doesn't satisfies

Riya Gupta - 4 years, 6 months ago

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if we take negative value,we get complex roots to the equation x^2-5x+16=0 which contradicts to question,since it is a natural number.

Sri priya Prerna - 4 years, 6 months ago

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No I'm not talking about the negative value of the root but the negative value of square root since every positive number has two opposite roots [ ( \sqrt{1} ) = ± 1 and ( \sqrt{16} ) = ± 4 ]

Rushikesh Jogdand - 4 years, 6 months ago

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-4 does not satisfy the initial conditions of the question. The question clearly states we need only the square of a natural number. -4 is not a natural number. Hence, it is illogical, subject to the initial conditions of the question, to write the square root of 16 as -4.

Aditya Parson - 4 years, 6 months ago

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@Aditya Parson It doesn't says square root must be natural. It says the numbers should be natural...

Rushikesh Jogdand - 4 years, 6 months ago

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@Aditya Parson No. You are again misunderstanding my side. I am telling that there is confusion between square root os the solutions. The solutions are 16 and 1 (which are natural numbers) see the two ways...

Take the solution 16[it is a natural OK]

according to condition-

x = 3√x + 4

LHS = 16

RHS = 3√16 + 4

but there are two values of √16

what to do about them????????

Rushikesh Jogdand - 4 years, 6 months ago

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@Rushikesh Jogdand You take the values and separately solve them. One will satisfy the condition the other wont, simple.

Aditya Parson - 4 years, 6 months ago

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\(\sqrt{16}\) = +4 and not equal to -4.

consider \(y^{2}\) = 16

y = +\(\sqrt{16}\) or y = -\(\sqrt{16}\)

So,

\(\sqrt{16}\) = 4 and - \(\sqrt{16}\) = -4

Furthermore, for all real x, \(\sqrt{x}\) is always positive. It is not a function that takes two values.

So + \(\sqrt{16}\) can be only one number and a positive number.

This is also the reason why * \(\sqrt{x^2}\) = | x| *

Samarth M.O. - 3 years, 11 months ago

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