To determine the possible values of b given the complex number x + 3bi and the equation x² = 13, let’s break it down step by step.
1. **Solve for x:** Given that x² = 13, we can find x by taking the square root of both sides. This gives us:
- x = √13 or
- x = -√13
2. **Substituting in the complex number:** Since we have two possible values for x, we can evaluate the corresponding complex numbers:
- If x = √13, the complex number becomes √13 + 3bi.
- If x = -√13, the complex number becomes -√13 + 3bi.
3. **Considering the value of b:** In this context, the value of b can be any real number. The presence of bi indicates that the complex number has an imaginary component, and b can vary to represent any point along the imaginary axis related to the real numbers defined by x.
In summary, since b is independent of the value of x, any real number would be a possible value for b. Thus, b can be any real number.