(x-4)2+8=0

Jasonbradley

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Jun 17, 2024

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Solving the Quadratic Equation: (x-4)² + 8 = 0

This article will guide you through the steps of solving the quadratic equation (x-4)² + 8 = 0.

Understanding the Equation

The equation (x-4)² + 8 = 0 represents a quadratic equation in standard form, which is ax² + bx + c = 0. In this case:

• a = 1 (coefficient of x²)
• b = -8 (coefficient of x)
• c = 24 (constant term)

Solving for x

We can solve for x using the following steps:

1. Expand the square: (x-4)² = x² - 8x + 16
2. Substitute: x² - 8x + 16 + 8 = 0
3. Simplify: x² - 8x + 24 = 0
4. Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
   • Substitute the values of a, b, and c:
     • x = (8 ± √((-8)² - 4 * 1 * 24)) / (2 * 1)
   • Simplify:
     • x = (8 ± √(-32)) / 2
     • x = (8 ± 4√(-2)) / 2
     • x = 4 ± 2√(-2)
5. Express in terms of imaginary unit: √(-2) = √(-1 * 2) = √(-1) * √2 = i√2
6. Final Solution: x = 4 ± 2i√2

Conclusion

Therefore, the solutions to the quadratic equation (x-4)² + 8 = 0 are x = 4 + 2i√2 and x = 4 - 2i√2. These solutions are complex numbers, meaning they involve the imaginary unit 'i' (where i² = -1).