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Solving the Quadratic Equation: (x-1)^2 - 25 = 0
This article will guide you through the steps of solving the quadratic equation (x-1)^2 - 25 = 0. We will explore different methods and understand the concepts involved.
1. Using the Square Root Property
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Step 1: Isolate the squared term. Add 25 to both sides of the equation: (x-1)^2 = 25
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Step 2: Take the square root of both sides. Remember to consider both positive and negative roots: x-1 = ±5
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Step 3: Solve for x. Add 1 to both sides of the equation for each case: x = 1 + 5 or x = 1 - 5 x = 6 or x = -4
Therefore, the solutions to the equation (x-1)^2 - 25 = 0 are x = 6 and x = -4.
2. Expanding and Using the Quadratic Formula
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Step 1: Expand the squared term. (x-1)^2 = x^2 - 2x + 1 So the equation becomes: x^2 - 2x + 1 - 25 = 0
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Step 2: Simplify the equation. x^2 - 2x - 24 = 0
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Step 3: Apply the Quadratic Formula. The quadratic formula is used to solve equations of the form ax^2 + bx + c = 0: x = [-b ± √(b^2 - 4ac)] / 2a In this case, a = 1, b = -2, and c = -24.
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Step 4: Substitute the values into the quadratic formula and solve. x = [2 ± √((-2)^2 - 4 * 1 * -24)] / (2 * 1) x = [2 ± √(100)] / 2 x = (2 ± 10) / 2 x = 6 or x = -4
As you can see, both methods lead to the same solutions: x = 6 and x = -4.
Conclusion
Solving quadratic equations can be done through various methods. Understanding the square root property and the quadratic formula empowers you to find solutions efficiently. It's important to remember that quadratic equations can have two solutions, one solution, or no solutions.