Solving Equations with Absolute Values and Complex Numbers

Understanding how to solve equations involving absolute values, particularly in the context of complex numbers, is a fundamental skill in algebra. This article will guide you through solving the equation x^2 - 1 3^2 - 1 and exploring both real and complex solutions.

Simplifying the Equation

Let's start by simplifying the given equation:

Step 1: Simplify the Right Side

First, we calculate the right side of the equation:

32 - 1  9 - 1  8

The equation simplifies to:

x^2 - 1  8

Using Absolute Values

The equation x^2 - 1 8 can be interpreted as an absolute value equation. This can be split into two cases:

Case 1: x^2 - 1 8

Adding 1 to both sides:

x^2  9

Taking the square root of both sides:

x  3 or x  -3

Case 2: x^2 - 1 -8

Similarly, for the second case:

x^2 - 1  -8

Adding 1 to both sides:

x^2  -7

Since x^2 cannot be negative, there are no real solutions from this case.

Final Solutions

The only real solutions to the equation x^2 - 1 8 are:

x  3 and x  -3

Thus, the solutions are x 3 and x -3.

Exploring Complex Solutions

While we have found the real solutions, let's explore the possibility of complex solutions using the equation x^2 - 1 -8:

x^2 - 1  -8

Adding 1 to both sides:

x^2  -7

Now, taking the square root of both sides, we get:

x  isqrt{7}

Therefore, the complete set of solutions includes:

For real numbers, squares are positive, so we can drop the absolute values on both sides, leading to the solutions:

x  -3

If we consider complex numbers, another solution can be x isqrt{11}.

Conclusion

This article has detailed the process of solving the equation x^2 - 1 3^2 - 1, demonstrating both the real and complex solutions. Understanding these concepts is crucial for solving a wide range of algebraic equations and working effectively with complex numbers.