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Solving the Equation: (x+1)(2x-3)-3(x-2)=2(x-1)^2
This article aims to guide you through the process of solving the equation (x+1)(2x-3)-3(x-2)=2(x-1)^2. We will break down each step to ensure a clear understanding of the solution.
Step 1: Expanding the equation
Begin by expanding the equation, removing the parentheses and simplifying the expression:
- (x+1)(2x-3): Apply the FOIL method (First, Outer, Inner, Last) to multiply the terms.
- (x * 2x) + (x * -3) + (1 * 2x) + (1 * -3) = 2x² - x + 2x - 3 = 2x² + x - 3
- -3(x-2): Distribute the -3 across the terms inside the parentheses.
- -3 * x + (-3) * -2 = -3x + 6
- 2(x-1)²: Square the term (x-1) and then multiply by 2.
- 2 * (x-1)² = 2 * (x² - 2x + 1) = 2x² - 4x + 2
Now, the equation becomes: 2x² + x - 3 - 3x + 6 = 2x² - 4x + 2
Step 2: Simplifying the equation
Combine like terms on both sides of the equation to simplify:
- 2x² - 2x² + x - 3x + 6 - 3 = -4x + 2
- -2x + 3 = -4x + 2
Step 3: Isolating the variable
Move all terms containing 'x' to one side of the equation and the constant terms to the other side.
- -2x + 4x = 2 - 3
- 2x = -1
Step 4: Solving for x
Finally, divide both sides by 2 to isolate 'x' and obtain the solution:
- x = -1/2
Conclusion
Therefore, the solution to the equation (x+1)(2x-3)-3(x-2)=2(x-1)² is x = -1/2.