3/x+2-(3/2-x)=2/x^2-4

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To solve the equation, we need to simplify it step by step:

3/x + 2 - (3/2 - x) = 2/x^2 - 4

First, simplify the fractions:

3/x + 2 - (3/2) + x = 2/x^2 - 4

Next, combine like terms:

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

Now, get a common denominator and combine terms:

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

Now, multiply both sides by x to get rid of the denominator:

3x + x^2 - 3x/2 = 2 - 2x

Simplify further:

2x + x^2/2 = 2 - 2x

Multiply through by 2 to clear the fraction:

4x + x^2 = 4 - 4x

Rearrange the equation to set it equal to 0:

x^2 + 8x - 4 = 0

Now we have a quadratic equation which can be solved using the quadratic formula:

x = (-b ± √(b^2 - 4ac))/2a

Plugging in the values for a, b, and c:

x = (-8 ± √(64 + 16))/2
x = (-8 ± √80)/2
x = (-8 ± 4√5)/2
x = -4 ± 2√5

Therefore, the solutions to the equation are:

x = -4 + 2√5
x = -4 - 2√5