If (x + 1/x) = 7, x 0. Find the positive value of (x - 1/x).

Question

If (x + 1/x) = 7, x > 0. Find the positive value of (x - 1/x).

Accepted Answer

Given: x + 1/x = 7, x > 0.

We need to find the positive value of (x - 1/x).

Step 1: Square both sides of the given equation:
(x + 1/x)^2 = 7^2  
=> x^2 + 2 + 1/x^2 = 49

Step 2: Rearrange to find x^2 + 1/x^2:  
=> x^2 + 1/x^2 = 49 - 2 = 47

Step 3: Now, consider (x - 1/x)^2:  
(x - 1/x)^2 = x^2 - 2 + 1/x^2  
= (x^2 + 1/x^2) - 2  
= 47 - 2 = 45

Step 4: Take the positive square root (since x > 0 and we want positive value):  
(x - 1/x) = √45 = √(9 * 5) = 3√5

Check options:  
Option 3 is 3√5, which matches our result.

However, the options given are:  
1. 5√4 = 5 * 2 = 10  
2. 5√5 = 5 * 2.236 = 11.18 approx  
3. 3√5 = 3 * 2.236 = 6.708 approx  
4. 4√5 = 4 * 2.236 = 8.944 approx

Our answer is 3√5, which corresponds to option 3.

Therefore, the positive value of (x - 1/x) is 3√5.

Answer

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