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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Thu Jul 26, 2012 9:43 pm

Guys, please help me with below:-

If x > 49, simplify x-5square root x -14/square root x - 7

Answer is square root x + 2

Thank you in advance.

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Source: Beat The GMAT — Problem Solving | Thu Jul 26, 2012 9:43 pm

shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Fri Jul 27, 2012 7:46 am

Can anyone help, please? I'm really desperate to solve this question.

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Fri Jul 27, 2012 7:54 am

shanice wrote:Can anyone help, please? I'm really desperate to solve this question.

Hi Shanice:

Its hard to read the question. Could you either rewrite it with brackets to make it clear, or perhaps, post a screenshot as an attachment? I will help.

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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Fri Jul 27, 2012 8:30 pm

Sorry about that. I'm not really familiar with the keyboard functions.

Anyway, I've re-typed the question with some help. Hope it's clear now.

1)If x>49, simplify x-5âˆšx-14 /âˆšx-7.

Answer is âˆšx+2

Note: The square root is for x only.

2)If x+1/x=10, then what is the value of x^2+1/x^2?

Answer is 98.

I hope you could help me with the two questions. I really appreciate your help.

A million thanks, eagleeye.

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sat Jul 28, 2012 2:54 am

shanice wrote:Sorry about that. I'm not really familiar with the keyboard functions.

Anyway, I've re-typed the question with some help. Hope it's clear now.

1)If x>49, simplify x-5âˆšx-14 /âˆšx-7.

Answer is âˆšx+2

Note: The square root is for x only.

2)If x+1/x=10, then what is the value of x^2+1/x^2?

Answer is 98.

I hope you could help me with the two questions. I really appreciate your help.

A million thanks, eagleeye.

You're welcome.

1. 1)(x-5âˆšx-14) /(âˆšx-7)

To make the question easier to solve, since x is under sqrt, let sqrt(x) = a.
Then, x = a^2.

Now, the expression becomes
(a^2-5a-14)/(a-7)
= (a^2-5a-2a+2a-14)/(a-7). (adding and subtracting 2a to factorize the expression)
= (a^2-7a +2a-14)/(a-7)
= ( a(a-7) +2(a-7) )/(a-7)
= ( (a+2)(a-7))/(a-7). (a-7 cancels out)
= a+2.

Since sqrt(x) =a
Final ans = a+2 = sqrt(x) + 2

2)If x+1/x=10, then what is the value of x^2+1/x^2?

x+1/x=10
(x+1/x)^2=10^2. (squaring both sides)
x^2 + (1/x)^2 + 2*x*(1/x) = 100. ( using (a+b)^2 = a^2 + b^2 + 2*a*b. here a= x, b= 1/x )
x^2 + (1/x)^2 + 2 = 100. (x*1/x =1, because x cancels out top and bottom)
x^2 + 1/x^2 = 100-2 = 98.

Cheers !

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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Sat Jul 28, 2012 9:39 pm

Hi eagleeye,

In reference to question 2, why do I need to do squaring? I don't understand that part.

Thank you for your help.

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sun Jul 29, 2012 6:04 am

shanice wrote:Hi eagleeye,

In reference to question 2, why do I need to do squaring? I don't understand that part.

Thank you for your help.

Because the question asks the value of x^2 + 1/x^2 (sum of two squares), its the logical thing to try to square x+1\x

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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Mon Jul 30, 2012 8:08 pm

Thank you very much, eagleeye. I understand it now and because of that I get to solve a lot of similar questions correctly.

I'm truly grateful.

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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Fri Aug 10, 2012 12:24 am

Hi eagleeye,

I'm having problem with one of the questions that you solved for me on 28th July.It's question 1.Below is the question together with your workings:-

1)(x-5âˆšx-14) /(âˆšx-7)

To make the question easier to solve, since x is under sqrt, let sqrt(x) = a.
Then, x = a^2.

Now, the expression becomes
(a^2-5a-14)/(a-7)
= (a^2-5a-2a+2a-14)/(a-7). (adding and subtracting 2a to factorize the expression)
= (a^2-7a +2a-14)/(a-7)
= ( a(a-7) +2(a-7) )/(a-7)
= ( (a+2)(a-7))/(a-7). (a-7 cancels out)
= a+2.

Since sqrt(x) =a
Final ans = a+2 = sqrt(x) + 2

My questions :- 1)In order to eliminate the square root, I need to apply squaring into the problem,
right?

So, a^2-5(âˆšx)^2-14 / (âˆšx)^2-7 Am I right?
Then, it becomes (a^2-5a-14)/(a-7)

Why you don't square the numerator - 5 & 14 and denominator - 7?

2)Where did you get -2a+2a from?

(a^2-5a-2a+2a-14)/(a-7).

I'm extremely sorry for bothering you with some of my silly questions but I came across this type of question again and got stuck.I think I may not know some of the concepts of eliminating square roots. Please use question 1 as an example.

Thank you, eagleeye.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Aug 10, 2012 1:48 am

shanice wrote:)If x>49, simplify x - 5âˆšx - 14 / âˆšx-7.

Answer is [spoiler]âˆšx+2[/spoiler]

Let x=64.
(x - 5âˆšx - 14) / (âˆšx-7) = (64 - 5âˆš64 - 14) / (âˆš64 - 7) = (64 - 5*8 - 14) / (8-7) = 10/1 = 10. This is our target.
Now we plug x=64 into the answers to see which yields our target of 10.
[spoiler]âˆšx+2[/spoiler] = âˆš64 + 2 = 8+2 = 10.

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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Fri Aug 10, 2012 7:27 am

Hi GMATGuruNY,

Thank you very much, Sir. I understand the usage of x>49 in this question now. That's an easier way to solve the question.

By the way, without using "x > 49" how can I eliminate the square root?

Thank you in advance.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Aug 10, 2012 8:25 am

shanice wrote:Hi GMATGuruNY,

Thank you very much, Sir. I understand the usage of x>49 in this question now. That's an easier way to solve the question.

By the way, without using "x > 49" how can I eliminate the square root?

Thank you in advance.

We can plug in any nonnegative value for x other than 49.
(If x=49, then the denominator = âˆšx - 7 = âˆš49 - 7 = 0, making the expression undefined.)

If x = 4, then:
(x - 5âˆšx - 14) / (âˆšx - 7) = (4 - 5âˆš4 - 14) / (âˆš4 - 7) = -20/-5 = 4.
Here, our target is 4.
Now we plug x=4 into the answers to see which yield our target of 4:
âˆšx + 2 = âˆš4 + 2 = 4.
Success!

Here's an alternate approach:

(x - 5âˆšx - 14) / (âˆšx - 7) = ?

In this sort of problem, the denominator is invariably a factor of the numerator.
Thus:
(âˆšx - 7)(a + b) = x - 5âˆšx - 14.

Note the values in red.
Since (âˆšx)(a) = x, we know that a = âˆšx:
(âˆšx - 7)(âˆšx + b) = x - 5âˆšx - 14.

Note the values in red.
Since (-7)(b) = -14, we know that b=2:
(âˆšx - 7)(âˆšx + 2) = x - 5âˆšx - 14.

A quick check confirms that the lefthand side is equal to the righthand side.

Thus:
(x - 5âˆšx - 14) / (âˆšx - 7) = âˆšx + 2.

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jaiswalamrita: Senior | Next Rank: 100 Posts; Posts: 34; Joined: Tue Apr 24, 2012 1:58 am

by jaiswalamrita » Fri Aug 10, 2012 10:48 am

GMATGuruNY wrote:
shanice wrote:)If x>49, simplify x - 5âˆšx - 14 / âˆšx-7.

Answer is [spoiler]âˆšx+2[/spoiler]

Let x=64.
(x - 5âˆšx - 14) / (âˆšx-7) = (64 - 5âˆš64 - 14) / (âˆš64 - 7) = (64 - 5*8 - 14) / (8-7) = 10/1 = 10. This is our target.
Now we plug x=64 into the answers to see which yields our target of 10.
[spoiler]âˆšx+2[/spoiler] = âˆš64 + 2 = 8+2 = 10.

Hi Mitch,

I'm really overwhelmed with your way of solving any problem. The way you use back solving for any and every question is amazing. I actively look for your explanation for any problem I get stuck.

A BIG thanks for being an online support for all GMAT aspirants.

Amrita

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shanice: Master | Next Rank: 500 Posts; Posts: 110; Joined: Thu Apr 05, 2012 8:48 amFollowed by: 1 member; Thanked: 3 times

by shanice » Fri Aug 10, 2012 10:42 pm

Got it. Thank you very much.

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