(x-4)^2-9=0

Jasonbradley

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Jun 17, 2024

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Solving the Quadratic Equation: (x-4)² - 9 = 0

This article will guide you through the process of solving the quadratic equation (x-4)² - 9 = 0. We'll explore the various methods for finding the solutions, also known as roots, of this equation.

Understanding the Equation

The equation (x-4)² - 9 = 0 is a quadratic equation because it contains a term with x² (after expanding the equation). It is in a simplified form, where the left side is a perfect square trinomial minus a constant.

Solving by Factoring

1. Recognize the difference of squares pattern: The equation can be rewritten as (x-4)² - 3² = 0. This is in the form of a² - b² = 0, which factors into (a+b)(a-b) = 0.
2. Apply the pattern: Substitute (x-4) for 'a' and 3 for 'b'. This gives us [(x-4) + 3][(x-4) - 3] = 0.
3. Solve for x: Expanding the brackets gives us (x-1)(x-7) = 0. Setting each factor to zero, we get x-1 = 0 or x-7 = 0. Solving these equations gives us x = 1 and x = 7.

Solving by the Square Root Property

1. Isolate the squared term: Add 9 to both sides of the equation: (x-4)² = 9.
2. Take the square root of both sides: √(x-4)² = ±√9. Remember to consider both positive and negative square roots.
3. Solve for x: This gives us x-4 = ±3. Adding 4 to both sides results in x = 4 ± 3.
4. Find the solutions: Therefore, the solutions are x = 1 and x = 7.

Solving using the Quadratic Formula

While the quadratic formula can be used for any quadratic equation, it is a less efficient method for this specific equation. However, we'll outline the steps for completeness:

1. Rewrite the equation in standard form: Expand the equation: x² - 8x + 16 - 9 = 0, simplifying to x² - 8x + 7 = 0.
2. Identify coefficients: In the standard quadratic equation ax² + bx + c = 0, we have a = 1, b = -8, and c = 7.
3. Apply the quadratic formula: The formula states: x = (-b ± √(b² - 4ac)) / 2a. Substitute the coefficients to find the solutions.
4. Simplify and solve: After calculations, you'll again arrive at x = 1 and x = 7.

Conclusion

We have successfully solved the equation (x-4)² - 9 = 0 using three different methods. All three methods lead to the same solutions, x = 1 and x = 7. Choosing the appropriate method depends on the specific form of the equation and personal preference. Understanding these different approaches will equip you to confidently tackle a wide range of quadratic equations.