%5E2%3D-3.jpeg "(k-4)^2=-3")
Solving the Equation: (k-4)^2 = -3
This equation presents a unique challenge because it involves a squared term equaling a negative number. Let's break down how to solve it and understand the implications.
Understanding the Basics
- Squaring a number always results in a non-negative value. This means that any real number multiplied by itself will always be zero or positive.
- Therefore, a perfect square cannot equal a negative number.
Solving the Equation
Given the equation (k-4)^2 = -3, we know that there are no real solutions for the variable 'k'. This is because the left side of the equation can never be negative, while the right side is fixed at -3.
Exploring Complex Numbers
To find solutions to this equation, we need to delve into the realm of complex numbers. Complex numbers are an extension of real numbers, including the imaginary unit 'i', where i² = -1.
Let's rewrite our equation:
(k-4)² = -3
Taking the square root of both sides:
k - 4 = ±√(-3)
Simplifying:
k - 4 = ±√3 * √(-1)
k - 4 = ±√3 * i
Solving for 'k':
k = 4 ± √3 * i
The Solution
Therefore, the solutions to the equation (k-4)² = -3 are complex numbers, specifically:
- k = 4 + √3 * i
- k = 4 - √3 * i
Conclusion
While the equation (k-4)² = -3 does not have real number solutions, it does have two complex solutions. This demonstrates that the realm of complex numbers allows us to solve equations that would be impossible to solve within the set of real numbers.