Final answer:
To find the unique values of x for the equation (x-1)(x-2)=3x-7, we expand, set the equation to zero, factor it, and then solve for x, resulting in one unique solution: x = 3.
Step-by-step explanation:
When solving the equation (x-1)(x-2)=3x-7, we are looking for the unique values of x that satisfy it. We can start by expanding the left side to get x² - 3x + 2. Then, we want to set the equation to zero by bringing all terms to one side, resulting in x² - 6x + 9 = 0. This equation factors further into (x - 3)² = 0, indicating that there is one unique solution x = 3.
To find the solution, follow these steps:
- Expand the left side: (x - 1)(x - 2) = x² - 3x + 2.
- Set the equation to zero: x² - 6x + 9 = 0.
- Factor the quadratic equation: (x - 3)² = 0.
- Solve for x: x = 3.