Question

Solve the following quadratic equation:

`(x - 1)/(2x + 1) + (2x - 1)/(x - 1) = 2, x ≠ -1/2, 1`

Answer 1

Given: `(x - 1)/(2x + 1) + (2x - 1)/(x - 1) = 2, x ≠ -1/2, 1`

Step-wise calculation:

1. Combine the left-hand side over the common denominator (2x + 1)(x – 1):

(x – 1)² + (2x – 1)(2x + 1) = 2(2x + 1)(x – 1)

2. Expand the numerators:

(x – 1)² = x² – 2x + 1

(2x – 1)(2x + 1) = 4x² – 1

Sum = x² – 2x + 1 + 4x² – 1 = 5x² – 2x

So, `(5x^2 - 2x)/((2x + 1)(x - 1)) = 2`.

3. Clear the denominator allowed since `x ≠ -1/2, 1`: 

5x² – 2x = 2(2x + 1)(x – 1)

4. Expand the right side:

2(2x + 1)(x – 1) = 2(2x² – x – 1) 

= 4x² – 2x – 2

5. Move all terms to one side:

5x² – 2x – (4x² – 2x – 2) = 0

⇒ x² + 2 = 0

6. Solve for x: 

x² = –2 

⇒ x = `±  isqrt(2)`

The equation has no real solutions; the solutions are x = `isqrt(2)` and x = `-isqrt(2)` both allowed since they do not equal the excluded real values `-1/2` or 1.