Problem Solving

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.

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

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.

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-5√x-14 /√x-7.

Answer is √x+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.

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-5√x-14 /√x-7.

Answer is √x+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-5√x-14) /(√x-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 !

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.

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

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.

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-5√x-14) /(√x-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(√x)^2-14 / (√x)^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.

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

Answer is [spoiler]√x+2[/spoiler]

Let x=64.
(x - 5√x - 14) / (√x-7) = (64 - 5√64 - 14) / (√64 - 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]√x+2[/spoiler] = √64 + 2 = 8+2 = 10.

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.

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 = √x - 7 = √49 - 7 = 0, making the expression undefined.)

If x = 4, then:
(x - 5√x - 14) / (√x - 7) = (4 - 5√4 - 14) / (√4 - 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:
√x + 2 = √4 + 2 = 4.
Success!

Here's an alternate approach:

(x - 5√x - 14) / (√x - 7) = ?

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

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

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

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

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

GMATGuruNY wrote:

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

Answer is [spoiler]√x+2[/spoiler]

Let x=64.
(x - 5√x - 14) / (√x-7) = (64 - 5√64 - 14) / (√64 - 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]√x+2[/spoiler] = √64 + 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

Got it. Thank you very much.