X-3+6/x+3 - x-3/x = 3/32

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To solve this equation, we need to first combine like terms and simplify the expression on both sides of the equation.

Given equation: x - 3 + 6/(x + 3) - (x - 3)/x = 3/32

Combining like terms in the equation, we get:

x - 3 + 6/(x + 3) - (x - 3)/x = 3/32
= x - 3 + 6/(x + 3) - (x - 3)/x
= x - 3 + 6/(x + 3) - (x - 3)(x + 3)/(x(x + 3))
= x - 3 + 6/(x + 3) - (x^2 - 9)/(x(x + 3))
= x - 3 + 6/(x + 3) - (x^2 - 9)/(x^2 + 3x)

Now, we can simplify the equation by finding a common denominator and combining the fractions:

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

Simplifying further:

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

Now, the equation becomes:

(-2x + 6)/(x(x + 3)) = 3/32

To solve this equation, cross multiply:

32(-2x + 6) = 3(x)(x + 3)

-64x + 192 = 3x^2 + 9x
3x^2 + 9x + 64x - 192 = 0
3x^2 + 73x - 192 = 0

Now, we have a quadratic equation 3x^2 + 73x - 192 = 0. We can solve this using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Where a = 3, b = 73, and c = -192.

Plugging these values into the formula gives:

x = (-73 ± √(73^2 - 43(-192))) / (2*3)
x = (-73 ± √(5329 + 2304)) / 6
x = (-73 ± √7633) / 6

Therefore, the solutions for x are:

x = (-73 + √7633) / 6
x = (-73 - √7633) / 6