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Intern  Joined: 26 Dec 2015
Posts: 3
If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (01:41) correct 37% (02:18) wrong based on 265 sessions

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If $$x^2 + \frac{9}{x^2} = 31$$, what is the value of $$x - \frac{3}{x}$$?

A. 36
B. 25
C. 9
D. 5
E. 3

Originally posted by maxx1234 on 28 Dec 2015, 00:53.
Last edited by Bunuel on 03 Oct 2017, 05:15, edited 2 times in total.
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Joined: 20 Jan 2015
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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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2
option D.

To find : x-3/x. Let it be t.
=> x-3/x = t
=> (x^2 + 9/x^2) - 2*x*3/x = t^2 (Squaring both sides).
=> (31) - 2*3 = 25
=> t^2 = 25. Thus t=5 or t=-5.

=> option D.
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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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Dont get this part

- 2*x*3/x
Current Student B
Joined: 20 Jan 2015
Posts: 47
Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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maxx1234 wrote:
Dont get this part

- 2*x*3/x

Have just used the formula (a-b)^2 = a^2 + b^2 - 2*a*b.
Here a is x and b is 3/x.

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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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ritikk13 wrote:
maxx1234 wrote:
Dont get this part

- 2*x*3/x

Have just used the formula (a-b)^2 = a^2 + b^2 - 2*a*b.
Here a is x and b is 3/x.

Consider Kudos if this helps. Hi ritikk,

if x-3/x is squared then resultant equation after substitution will be 31 - (6/x). How do you arrive at 5 from this when x is still present?
Math Expert V
Joined: 02 Sep 2009
Posts: 58163
Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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Senthil7 wrote:
ritikk13 wrote:
maxx1234 wrote:
If x^2 + 9/x^2 = 31, what is the value of x - 3/x

A 36
B 25
C 9
D 5
E 3

Dont get this part

- 2*x*3/x

Have just used the formula (a-b)^2 = a^2 + b^2 - 2*a*b.
Here a is x and b is 3/x.

Consider Kudos if this helps. Hi ritikk,

if x-3/x is squared then resultant equation after substitution will be 31 - (6/x). How do you arrive at 5 from this when x is still present?

$$(x-\frac{3}{x})^2=x^2-2*x*\frac{3}{x}+(\frac{3}{x})^2=x^2-6+\frac{9}{x^2}$$

Since $$x^2 + \frac{9}{x^2} = 31$$, then $$(x-\frac{3}{x})^2=x^2-6+\frac{9}{x^2}=31-6=25$$.

Finally, $$(x-\frac{3}{x})^2=25$$ --> $$x-\frac{3}{x}=5$$ or -5.

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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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very nice question but needs smart eyes to catch the idea.
Are there any alternate approach ?
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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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Bunuel thanks a lot. The misunderstanding is due to the font type not sure if it was x-3 the whole term by x or just x minus 3/x. That issue with members posting can solve the issue i think any way out or any suggestion like you have typed now appropriately? Bunuel
Math Expert V
Joined: 02 Sep 2009
Posts: 58163
Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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Senthil7 wrote:
Bunuel thanks a lot. The misunderstanding is due to the font type not sure if it was x-3 the whole term by x or just x minus 3/x. That issue with members posting can solve the issue i think any way out or any suggestion like you have typed now appropriately? Bunuel

Mathematically x - 3/x can only mean x minus 3/x. If it were x-3 over x it would e written as (x - 3)/x.
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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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1
1
maxx1234 wrote:
If $$x^2 + \frac{9}{x^2} = 31$$, what is the value of $$x - \frac{3}{x}$$?

A. 36
B. 25
C. 9
D. 5
E. 3

Let’s square x - 3/x first:

(x - 3/x)^2 = x^2 - 6 + 9/x^2

Since x^2 + 9/x^2 = 31, we have x^2 - 6 + 9/x^2 = 31 - 6 = 25.

So, (x - 3/x)^2 = 25 and:

x - 3/x = ±5

We see that only 5 is among the answer choices.

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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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ScottTargetTestPrep wrote:
maxx1234 wrote:
If $$x^2 + \frac{9}{x^2} = 31$$, what is the value of $$x - \frac{3}{x}$$?

A. 36
B. 25
C. 9
D. 5
E. 3

Let’s square x - 3/x first:

(x - 3/x)^2 = x^2 - 6 + 9/x^2

Since x^2 + 9/x^2 = 31, we have x^2 - 6 + 9/x^2 = 31 - 6 = 25.

So, (x - 3/x)^2 = 25 and:

x - 3/x = ±5
We see that only 5 is among the answer choices.

Hey pushpitkc,

can you shed some light on the solution above ..i dont get the steps (highlighted) after this "Since x^2 + 9/x^2 = 31"

many thanks Senior Manager  G
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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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1
dave13

You can convert any a^2+b^2 to a^2+b^2-2ab+2ab=(a-b)^2+2ab
x^4 + 9/x^2 = x^4 + 9/x^2 - 2*(x^2 *3/x^2) + 2*(x^2 *3/x^2)
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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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1
dave13 wrote:
ScottTargetTestPrep wrote:
maxx1234 wrote:
If $$x^2 + \frac{9}{x^2} = 31$$, what is the value of $$x - \frac{3}{x}$$?

A. 36
B. 25
C. 9
D. 5
E. 3

Let’s square x - 3/x first:

(x - 3/x)^2 = x^2 - 6 + 9/x^2

Since x^2 + 9/x^2 = 31, we have x^2 - 6 + 9/x^2 = 31 - 6 = 25.

So, (x - 3/x)^2 = 25 and:

x - 3/x = ±5
We see that only 5 is among the answer choices.

Hey pushpitkc,

can you shed some light on the solution above ..i dont get the steps (highlighted) after this "Since x^2 + 9/x^2 = 31"

many thanks Hey dave13

We are given $$x^2 + \frac{9}{x^2} = 31$$ in the queston stem -> Lets call this equation (1)

Once, we have squared the equation, $$(x - \frac{3}{x})^2$$ to get $$x^2 - 6 + \frac{9}{x^2}$$

After re-arranging $$(x - \frac{3}{x})^2 = x^2 + \frac{9}{x^2} - 6 = 31 - 6 = 25$$ [from equation (1)]

Now, $$(x - \frac{3}{x})^2 = 25$$ -> $$(x - \frac{3}{x}) = \sqrt{25} = +5$$ or $$-5$$

Therefore, the answer option which matches the value of $$(x - \frac{3}{x})$$ is 5(Option D)

Hope this helps you
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Manager  B
Joined: 17 Jul 2016
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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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One could try and substitute values.

if x=1 the expression (x^2+9/x^2) = 10
if x=2, 4+(9/4)=6.25
if x=3, 10
if x=4, 16&9/16
if x=5, 25&9/25
if x=6 36&1/4

We want the expression to =31 so x=5.5 is a good estimate.

x-3/x=

5.5-(3/5.5)=

5.5 - (little more than .5)=close to 5
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Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?  [#permalink]

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maxx1234 wrote:
If $$x^2 + \frac{9}{x^2} = 31$$, what is the value of $$x - \frac{3}{x}$$?

A. 36
B. 25
C. 9
D. 5
E. 3

Let $$a$$ = the correct answer.

$$x - \frac{3}{x}=a$$

Since $$(m-n)^2 = m^2 + n^2 - 2mn$$, squaring both sides yields the following:
$$(x - \frac{3}{x})^2=a^2$$

$$x^2 - \frac{9}{x^2} - 2(x)(\frac{3}{x})=a^2$$

$$x^2 - \frac{9}{x^2} - 6=a^2$$

Substituting $$x^2 + \frac{9}{x^2} = 31$$ into the resulting blue equation, we get:
$$31 - 6=a^2$$

$$25 = a^2$$

±$$5=a$$

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Available for tutoring in NYC and long-distance. Re: If x^2 + 9/x^2 = 31, what is the value of x - 3/x?   [#permalink] 22 Mar 2019, 10:42
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