If x^2 + 9/x^2 = 31, what is the value of x - 3/x?

Question

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

Bunuel · Accepted Answer

(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.

Answer: D.

BrentGMATPrepNow · Answer

Let  k = x - \frac{3}{x}

Square both sides to get:  k^2 = (x - \frac{3}{x})^2

Use FOIL to expand and simplify the right side:  k^2 = x^2 - 6 + \frac{9}{x^2}

Rewrite the right side as follows:  k^2 = (x^2 + \frac{9}{x^2}) - 6

Since it's given that  x^2 + \frac{9}{x^2} = 31 , we can substitute as follows:  k^2 = (31) - 6

Simplify:  k^2 = 25

So,  k = 5  or  k = -5

Check the answer choices.... The correct answer must be D

WaitingSurprises · Answer

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.

maxx1234 · Answer

Dont get this part

- 2*x*3/x

WaitingSurprises · Answer

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.

Senthil7 · Answer

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?

hatemnag · Answer

very nice question but needs smart eyes to catch the idea. 
Are there any alternate approach ?

Senthil7 · Answer

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

Bunuel · Answer

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.

ScottTargetTestPrep · Answer

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.

Answer: D

dave13 · Answer

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

Hero8888 · Answer

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)

pushpitkc · Answer

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

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

Henry S. Hudson Jr. · Answer

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

GMATGuruNY · Answer

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

D

kskumar · Answer

let y= 3/x => xy=3 ----- 1
x^2+y^2= 31 --- 2
x - y = ?

(x-y)^2 = x^2 + y^2- 2xy ---- 3
subs 1 and 2 in 3
we get
 (x-y)^2 = 31-6 = 25 = 5^2

so x-y = 5

CrackverbalGMAT · Answer

Simple logic and a basic understanding of Algebraic identities will tell you that x –  \frac{3}{x } cannot be as big a number as 36.

If x –  \frac{3}{x}  = 36,  x^2  +  \frac{9}{x^2}  will not be as small as 31 since squaring 36 will give us a large number like 1296.

What do I mean?

(x – \frac{3}{x}) ^2  =  x^2 + \frac{9}{x^2} – 2*x * \frac{3}{x}

Since (x- \frac{3}{x} ) = 36,  (36)^2  =  x^2  +  \frac{9 }{ x^2}  – 6 and therefore, x^2  +  \frac{9}{x^2}  cannot be as small as 31.

Answer options A and B can be easily eliminated. A little more deliberation will tell you that 9 also is big enough to be eliminated, so answer option C also can be ruled out.

On the other hand, 3 is too small a value for x –  \frac{3}{x}  which helps us eliminate answer option E.

The correct answer option is D.

If (x- \frac{3}{x} ) = 5,  (x-\frac{3}{x})^2  = 25.

Therefore,  x^2 + \frac{9}{x^2}  – 6 = 25 or  x^2 + \frac{9}{x^2}  = 31, as given in the question data.

Hope that helps!
Aravind B T

vibhorrokstar19 · Answer

Hi, I just solved this question using simple logic, the expression x^2+9/x^2 =31 tells us that x must be greater than 5 and less than 6. Also when we put maximum and min possible value for x in the expression we get the answer in the range of 4.ab < X <5.cd . Therefore answer must be within the range.

open for a discussion, let me know if there can be some exceptions to the above approach.

open for a discussion, let me know if there can be some exceptions to the above approach.

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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?

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If x^2 + 9/x^2 = 31, what is the value of x - 3/x?

Prep Toolkit

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