(x-2)%3D(x-2)(x%2B1).jpeg "(3x+1)(x-2)=(x-2)(x+1)")
Solving the Equation (3x+1)(x-2) = (x-2)(x+1)
This equation presents a straightforward example of solving a quadratic equation. Let's break down the steps:
1. Expanding the Equation
First, we need to expand both sides of the equation by multiplying the terms:
- Left side: (3x + 1)(x - 2) = 3x² - 6x + x - 2 = 3x² - 5x - 2
- Right side: (x - 2)(x + 1) = x² + x - 2x - 2 = x² - x - 2
Now the equation looks like this: 3x² - 5x - 2 = x² - x - 2
2. Simplifying the Equation
To solve for x, we need to bring all the terms to one side. Let's subtract x² and -x - 2 from both sides:
3x² - 5x - 2 - x² + x + 2 = 0
This simplifies to: 2x² - 4x = 0
3. Factoring the Equation
The equation can now be factored by taking out the greatest common factor, 2x:
2x(x - 2) = 0
4. Solving for x
For the product of two terms to equal zero, at least one of them must be zero. This gives us two possible solutions:
- 2x = 0 => x = 0
- x - 2 = 0 => x = 2
Conclusion
Therefore, the solutions to the equation (3x + 1)(x - 2) = (x - 2)(x + 1) are x = 0 and x = 2.