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Solving the Quadratic Equation: (x-3)^2 - 64 = 0
This article will guide you through solving the quadratic equation (x-3)^2 - 64 = 0. We'll explore different methods to find the solutions for 'x'.
Understanding the Equation
The equation (x-3)^2 - 64 = 0 is a quadratic equation because the highest power of 'x' is 2. It can be solved using various methods, and we'll demonstrate two popular approaches:
1. Using Square Roots:
- Simplify: Start by adding 64 to both sides of the equation: (x-3)^2 = 64
- Take the Square Root: Take the square root of both sides. Remember that there are two possible solutions: x - 3 = 8 or x - 3 = -8
- Solve for x: Add 3 to both sides in each equation: x = 11 or x = -5
2. Factoring:
- Recognize the Difference of Squares: The equation can be rewritten as (x-3)^2 - 8^2 = 0. This is in the form of a difference of squares: a^2 - b^2 = (a+b)(a-b)
- Factor: Apply the difference of squares pattern: (x-3 + 8)(x-3 - 8) = 0
- Solve for x: Set each factor equal to zero and solve: x + 5 = 0 or x - 11 = 0 x = -5 or x = 11
Solutions
Both methods lead to the same solutions: x = 11 and x = -5. These are the values of 'x' that satisfy the given quadratic equation.
Conclusion
Solving quadratic equations is a fundamental skill in algebra. Understanding different methods like using square roots and factoring allows you to find the solutions efficiently. In this case, both methods yielded the same solutions for the equation (x-3)^2 - 64 = 0, confirming their accuracy.