If x + 1/x = √40 and x 1, what is the value of x - 1/x ?

Question

If  x+\frac{1}{x}=√40  and  x>1 , what is the value of  x-\frac{1}{x} ?

A. 4
B. 5
C. 6
D. 24
E. 36

BrentGMATPrepNow · Accepted Answer

First recognize that  (x+\frac{1}{x})^2  and  (x-\frac{1}{x})^2  look very similar once expanded and simplified.

(x+\frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2}  
and 
 (x-\frac{1}{x})^2 = x^2 - 2 + \frac{1}{x^2}

With this in mind, here's my solution: 
Take:  x+\frac{1}{x}=√40 
Square both sides:  (x+\frac{1}{x})^2=(√40)^2 
Simplify:  x^2 + 2 + \frac{1}{x^2} = 40 
Subtract 4 from both sides of the equation:  x^2 - 2 + \frac{1}{x^2} = 36 
Factor the left side:  (x-\frac{1}{x})^2 = 36 
Take the square root of both sides:  x-\frac{1}{x} = 6

Answer: C

PyjamaScientist · Answer

Perhaps (C).

x + \frac{1}{x} = \sqrt{40} 
 \sqrt{40}  is between 6 and 7
Since x is greater than one and 1/x is a fraction. 
It  x - \frac{1}{x}  should be around 6 in my opinion.

Fdambro294 · Answer

The following Property holds true for all real values of A and B

(a + b)^2 — (a - b)^2 = +4(a)(b)

If you perform the algebra and carry the minus sign through the 2nd expression, you will see that it is valid.

In this problem

Let X = a

Let (1/X) = b

We are given that: X + (1/X) = sqrt(40)

Or

(a + b) = sqrt(40)

Square both sides and you have:

(a + b)^2 = 40

Now put this into our template at the top

(40) — (a - b)^2 = +4(a)(b)

if we take A and multiply it by B:

(a) (b) = (X) (1/X) = (X / X) = 1

So substitute in 1 for (a)(b)

40 — (a - b)^2 = 4(1)

Rearrange the equation

36 = (a - b)^2

Take the square root of a variable expression squared— 6 will be equal to the Absolute Value of (a - b)

= 6

And insert our terms

= 6

Since X > 1 ——-> the quantity inside the Modulus will always be a positive quantity. x will be some value greater than 1 and (1/X) will be a positive proper fraction between 0 and 1

We can therefore remove the Modulus because it is redundant and we get

X + (1/X) = 6

Answer
6

C

Posted from my mobile device

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If x + 1/x = √40 and x > 1, what is the value of x - 1/x ?

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