Find the roots of 6⁄x - 2⁄x - 1 - 1⁄x - 2 = 0 4

Explanation

To solve the equation: 6⁄ 2⁄ 1 x- x - 1 - ⁄x - 2 = 0 we can find a common denominator and simplify. Step 1: Find the common denominator The common denominator for 6⁄x, 2⁄x - 1, and 1⁄x - 2 is x(x - 1)(x - 2). Step 2: Rewrite the equation with the common denominator 6(x - 1)(x - 2) - 2x(x - 2) - x(x - 1)⁄ x(x - 1)(x - 2) = 0 Since the denominator cannot be zero, we only need to solve the numerator for zero: 6(x - 1)(x - 2) - 2x(x - 2) - x(x - 1) = 0 Step 3: Expand each term in the numerator Expanding each term: 1. 6(x - 1)(x - 2) = 6(x2 - 3x + 2) = 6x2 - 18x + 12 2. -2x(x - 2) = -2x2 + 4x 3. -x(x - 1) = -x2 + x Step 4: Combine like terms Now we combine these terms: (6x2 - 2x2 - x2) + (-18x + 4x + x) + 12 = 0 3x2 - 13x + 12 = 0 Step 5: Solve the quadratic equation Now we solve for x using the quadratic formula: 2 x = -(-13) ± √((-13) - 4 × 3 × 12)⁄2 × 3 x = 13 ± √(169 - 144)⁄6 x = 13 ± √25⁄6 x = 13 ± 5⁄6 So, we have two possible values for x: 1. x = 13 + 5⁄6 = 18⁄6 = 3 2. x = 13 - 5⁄6 = 8⁄6 = 4⁄3 Answer The roots of the equation are: x = 3 and x = 4⁄3