(x+3i)(x-3i)=34

2 min read Jun 16, 2024

Solving the Equation: (x + 3i)(x - 3i) = 34

This equation involves complex numbers and requires a bit of manipulation to solve for the unknown variable x. Let's break down the steps:

Understanding Complex Numbers

First, recall that complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as √-1.

Expanding the Equation

We can solve this equation by expanding the left-hand side using the difference of squares pattern:

(x + 3i)(x - 3i) = x² - (3i)²

Remembering that i² = -1, we can simplify further:

x² - (3i)² = x² - 9(-1) = x² + 9

Solving for x

Now our equation becomes:

x² + 9 = 34

Subtracting 9 from both sides:

x² = 25

Taking the square root of both sides (remembering both positive and negative roots):

x = ±5

Conclusion

Therefore, the solutions to the equation (x + 3i)(x - 3i) = 34 are x = 5 and x = -5.