If x is an integer such that x + 1/x = 65/8, then find the value of x - 1/x.

Question

Accepted Answer

Given: x + 1/x = 65/8

We need to find x - 1/x.

Step 1: Square both sides of the given equation:
(x + 1/x)^2 = (65/8)^2

Expanding the left side:
x^2 + 2 + 1/x^2 = 4225/64

Step 2: Rearrange to find x^2 + 1/x^2:
x^2 + 1/x^2 = 4225/64 - 2
= 4225/64 - 128/64
= (4225 - 128)/64
= 4097/64

Step 3: Use the identity:
(x - 1/x)^2 = x^2 - 2 + 1/x^2
= (x^2 + 1/x^2) - 2
= 4097/64 - 2
= 4097/64 - 128/64
= (4097 - 128)/64
= 3969/64

Step 4: Take the square root:
x - 1/x = ±√(3969/64) = ±(√3969)/8

Since 3969 = 63^2,

x - 1/x = ±63/8

Step 5: Determine the sign.
Given x + 1/x = 65/8 (positive), and x is an integer, x must be positive.

So, x - 1/x = 63/8.

Therefore, the correct option is 63/8.

Answer

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