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))/x = (-8 ± √80)/x = (-8 ± 4√5)/x = -4 ± 2√5
Therefore, the solutions to the equation are:
x = -4 + 2√x = -4 - 2√5
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))/
x = (-8 ± √80)/
x = (-8 ± 4√5)/
x = -4 ± 2√5
Therefore, the solutions to the equation are:
x = -4 + 2√
x = -4 - 2√5