(x%2B5i).jpeg "(x-5i)(x+5i)")
Understanding Complex Conjugates: (x-5i)(x+5i)
This expression involves the multiplication of two complex numbers that are complex conjugates of each other. Let's break down the concept and the solution:
What are Complex Conjugates?
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. For example, the complex conjugate of (a + bi) is (a - bi).
Key Property: When you multiply complex conjugates, the result is always a real number. This is because the imaginary terms cancel out.
Solving (x-5i)(x+5i)
Let's use the distributive property (or FOIL method) to expand the expression:
(x - 5i)(x + 5i) = x(x + 5i) - 5i(x + 5i)
Expanding further:
= x² + 5xi - 5xi - 25i²
Notice that the terms 5xi and -5xi cancel each other out. Remember that i² = -1. Therefore:
= x² - 25(-1)
= x² + 25
Conclusion
As you can see, multiplying (x-5i) and (x+5i), which are complex conjugates, results in a real number expression, x² + 25. This demonstrates the key property of complex conjugates, making them useful in various mathematical applications.