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Solving the Quadratic Equation: (x-1)^2 - 3x + 4 = 0
This article will guide you through the steps of solving the quadratic equation (x-1)^2 - 3x + 4 = 0. We will explore different methods and provide a comprehensive solution.
1. Expanding and Simplifying the Equation
First, we need to expand the squared term and simplify the equation:
(x-1)^2 - 3x + 4 = 0 x^2 - 2x + 1 - 3x + 4 = 0 x^2 - 5x + 5 = 0
2. Using the Quadratic Formula
The quadratic formula is a powerful tool for solving equations of the form ax^2 + bx + c = 0. In this case, a = 1, b = -5, and c = 5.
The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a
Substituting our values:
x = (5 ± √((-5)^2 - 4 * 1 * 5)) / 2 * 1 x = (5 ± √(25 - 20)) / 2 x = (5 ± √5) / 2
Therefore, the solutions to the equation are:
x = (5 + √5) / 2 x = (5 - √5) / 2
3. Conclusion
We have successfully solved the quadratic equation (x-1)^2 - 3x + 4 = 0 using the quadratic formula. The solutions are x = (5 + √5) / 2 and x = (5 - √5) / 2.
This demonstrates the utility of the quadratic formula in finding solutions to quadratic equations, even when they appear in a more complex form.