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If x-2=x+2 , x=?

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If x-2=x+2 , x=?  [#permalink]

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New post Updated on: 12 Jul 2017, 19:57
1
12
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

56% (02:09) correct 44% (02:19) wrong based on 94 sessions

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If \(x-2=\sqrt{x}+\sqrt{2}\) , x=?

A. \(3+2\sqrt{2}\)
B. \(3-2\sqrt{2}\)
C. \(3+\sqrt{2}\)
D. \(3-\sqrt{2}\)
E. \(2+2\sqrt{3}\)

Original Version wrote:
If \(x-2=\sqrt{x}+\sqrt{2}\) , x=?

A. \(3+2\sqrt{2}\)
B. \(3-2\sqrt{2}\)
C. \(3+\sqrt{2}\)
D. \(3-\sqrt{2}\)
E. \(2+2[fraction]3[/fraction]\)

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Originally posted by MathRevolution on 12 Jul 2017, 01:06.
Last edited by broall on 12 Jul 2017, 19:57, edited 2 times in total.
Fixed typo
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 12 Jul 2017, 05:47
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1
MathRevolution wrote:
If \(x-2=\sqrt{x}+\sqrt{2}\) , x=?

A. \(3+2\sqrt{2}\)
B. \(3-2\sqrt{2}\)
C. \(3+\sqrt{2}\)
D. \(3-\sqrt{2}\)
E. \(2+2[fraction]3[/fraction]\)


\(x-2=\sqrt{x}+\sqrt{2}\)
\((\sqrt{x}-\sqrt{2} )(\sqrt{x}+\sqrt{2}) = (\sqrt{x}+\sqrt{2})\)
\(\sqrt{x}-\sqrt{2}-1 = 0\)
\(\sqrt{x} = 1+\sqrt{2}\)
\(x = 3 +2\sqrt{2}\)

Answer A
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 12 Jul 2017, 01:55
MathRevolution

This is how I started solving this question but don't know how to solve it further..

x-sq.root x= 2+sq.root 2
Squaring both the sides

x^2+ x- 2x*sq.root x= 2+4+4*sq.root 2

x^2-x*sq.root x= 6+4* sq.root 2

x(x- sq.root x)= 2(3+2* sq.root 2)


Please help.
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 12 Jul 2017, 02:24
1
MathRevolution wrote:
If \(x-2=\sqrt{x}+\sqrt{2}\) , x=?

A. \(3+2\sqrt{2}\)
B. \(3-2\sqrt{2}\)
C. \(3+\sqrt{2}\)
D. \(3-\sqrt{2}\)
E. \(2+2[fraction]3[/fraction]\)


MathRevolution
Please correct the formatting for Option E. if you meant it to be \(2+2\sqrt{3}\)

As for the solution :

Since \(x-2=\sqrt{x}+\sqrt{2}\), we can rewrite the statement to be \(x - \sqrt{x} = 2 + \sqrt{2}\)
Because \(\sqrt{2}\) = 1.4(approximately) , the value of \(x - \sqrt{x}\) = 2 + \(\sqrt{2}\) = 2 +1.4 = 3.4

Using answer options,
Option A, if x = \(3+2\sqrt{2}\), the value of x = 3 + 2*1.4 = 3 + 2.8 = 5.8

Now using this value of x in the expression \(x - \sqrt{x}\),

\(x - \sqrt{x} = 5.8 - \sqrt{5.8} = 5.8 - 2.4 = 3.4\) because \(2.4^2 = 5.76\) which is closest to 5.8

Hence, making Option A our correct answer!
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 12 Jul 2017, 03:32
Solution given by pushpitkc is the best approach.

It can also be solved by plugin

\(X=3+2(2)ˆ1/2\)

\(3+2(2)ˆ1/2-2=(3+2(2)ˆ1/2)ˆ1/2+(2)ˆ1/2\)

\(1+2(2)ˆ1/2-(2)ˆ1/2=(3+2(2)ˆ1/2)ˆ1/2\)

\((1+(2)ˆ1/2)ˆ2=3+2(2)ˆ1/2\)-Squaring both the sides

\((1)ˆ2+2(2)ˆ1/2+2=3+2(2)ˆ1/2\)
\(3+2(2)ˆ1/2=3+2(2)ˆ1/2\)
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 12 Jul 2017, 05:07
pushpitkc wrote:
MathRevolution wrote:
If \(x-2=\sqrt{x}+\sqrt{2}\) , x=?

A. \(3+2\sqrt{2}\)
B. \(3-2\sqrt{2}\)
C. \(3+\sqrt{2}\)
D. \(3-\sqrt{2}\)
E. \(2+2[fraction]3[/fraction]\)


MathRevolution
Please correct the formatting for Option E. if you meant it to be \(2+2\sqrt{3}\)

As for the solution :

Since \(x-2=\sqrt{x}+\sqrt{2}\), we can rewrite the statement to be \(x - \sqrt{x} = 2 + \sqrt{2}\)
Because \(\sqrt{2}\) = 1.4(approximately) , the value of \(x - \sqrt{x}\) = 2 + \(\sqrt{2}\) = 2 +1.4 = 3.4

Using answer options,
Option A, if x = \(3+2\sqrt{2}\), the value of x = 3 + 2*1.4 = 3 + 2.8 = 5.8

Now using this value of x in the expression \(x - \sqrt{x}\),

\(x - \sqrt{x} = 5.8 - \sqrt{5.8} = 5.8 - 2.4 = 3.4\) because \(2.4^2 = 5.76\) which is closest to 5.8

Hence, making Option A our correct answer!




Great method.

Squaring both the sides is a very time consuming approach.

Also if there was some other value instead of sq.root 2, may be sq.root 7 or 11......How will we solve then? Should we learn the sq.roots?


Thanks
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 12 Jul 2017, 06:55
MathRevolution wrote:
If \(x-2=\sqrt{x}+\sqrt{2}\) , x=?

A. \(3+2\sqrt{2}\)
B. \(3-2\sqrt{2}\)
C. \(3+\sqrt{2}\)
D. \(3-\sqrt{2}\)
E. \(2+2[fraction]3[/fraction]\)



I Couldn't figure out the solution at first. But, later understood that this can be solved by rewriting the LHS as (a+b)*(a-b).

As pointed out in previous post. Good question- Thanks for posting.
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 14 Jul 2017, 00:05
==> You get x-2=√x+√2 , (√x-√2)(√x+√2)= √x+√2. If you divide √x+√2 on both sides, you get √x-√2=1, then √x=√2+1. Then, if you square both sides, you get x=(√2+1)^2=2+1+2√2 =3+2√2.

The answer is A.
Answer: A
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Re: If x-2=x+2 , x=?  [#permalink]

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New post 16 Jul 2017, 01:38
1
x-2 = √x+√2
Use conjugate for √x+√2 = √x-√2

x-2 = (√x+√2) (√x-√2)/(√x-√2)

x-2 = (x-2)/(√x-√2)

√x-√2 = 1

√x = 1+√2

x = (1+√2)^2

x = 3+2√2 ---- Option A
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Re: If x-2=x+2 , x=?  [#permalink]

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Re: If x-2=x+2 , x=? &nbs [#permalink] 22 Dec 2018, 01:18
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