(x-1)(x-2)+(x-2)(x-3)+(x-3)(x-1)=0

2 min read Jun 17, 2024

Solving the Equation: (x-1)(x-2) + (x-2)(x-3) + (x-3)(x-1) = 0

This equation presents a polynomial expression that we can simplify and solve for the unknown variable 'x'. Here's a step-by-step breakdown of the solution:

1. Expanding the Equation

First, we need to expand the products in the equation:

(x-1)(x-2) = x² - 3x + 2
(x-2)(x-3) = x² - 5x + 6
(x-3)(x-1) = x² - 4x + 3

Now, the equation becomes:

(x² - 3x + 2) + (x² - 5x + 6) + (x² - 4x + 3) = 0

2. Combining Like Terms

Next, we combine the terms with the same power of 'x':

(x² + x² + x²) + (-3x - 5x - 4x) + (2 + 6 + 3) = 0

This simplifies to:

3x² - 12x + 11 = 0

3. Solving the Quadratic Equation

We now have a quadratic equation in the form of ax² + bx + c = 0. There are multiple ways to solve this, but we'll use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 3, b = -12, and c = 11. Plugging these values into the formula:

x = (12 ± √((-12)² - 4 * 3 * 11)) / (2 * 3)

x = (12 ± √(144 - 132)) / 6

x = (12 ± √12) / 6

x = (12 ± 2√3) / 6

4. Simplifying the Solutions

Finally, we simplify the solutions:

x = (6 ± √3) / 3

Therefore, the solutions to the equation (x-1)(x-2) + (x-2)(x-3) + (x-3)(x-1) = 0 are:

x = (6 + √3) / 3
x = (6 - √3) / 3