Solve for x 1/(x-1)-2/(x^2)=0

Problem

1/(x−1)−2/(x2)=0

Solution

1. Identify the domain of the equation by noting that the denominators cannot be zero, which means x≠1 and x≠0

2. Move the second term to the right side of the equation to isolate the fractions.

1/(x−1)=2/(x2)

3. Cross-multiply to eliminate the fractions.

1⋅x2=2⋅(x−1)

4. Distribute the constant on the right side.

x2=2*x−2

5. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 2*x and adding 2 to both sides.

x2−2*x+2=0

6. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−2 and c=2

x=(−(−2)±√(,(−2)2−4*(1)*(2)))/(2*(1))

7. Simplify the expression under the square root (the discriminant).

x=(2±√(,4−8))/2

x=(2±√(,−4))/2

8. Express the result using the imaginary unit i where √(,−4)=2*i

x=(2±2*i)/2

9. Divide both terms in the numerator by 2 to find the final complex solutions.

x=1±i

Final Answer

1/(x−1)−2/(x2)=0⇒x=1±i

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