Reliable diagnosis and differentiation of bipolar disorder pose significant challenges due to symptom overlap with other psychiatric disorders. Inaccurate or delayed diagnosis can severely impact patient lifestyle and health, underscoring the need for precise diagnostic methods. A reliable diagnosis is crucial for effective treatment and interventions, helping patients return to routine life and offering economic and non-economic benefits. Therefore, accurate diagnostic approaches that minimize dependence on environmental or physical conditions and patient-reported symptoms are imperative. Electroencephalogram (EEG) signals, which provide significant diagnostic information, are commonly used in diagnosing psychiatric diseases. This study focuses on classifying bipolar disorder using advanced EEG signal processing and analysis. Recording EEG signals and extracting valuable features can be analyzed with intelligent methods to achieve high diagnostic accuracy. Specifically, our study employs Discrete Wavelet Transform (DWT) and Principal Component Analysis (PCA) for feature extraction, followed by a GoogLeNet-based approach optimized with Genetic Algorithms (GA). Our findings reveal that the GoogLeNet-based method achieves a superior accuracy of 99.1%, significantly outperforming conventional neural network approaches. Additionally, using the Brain Health Index (BHI) to represent brain activity in different frequency bands provides a clear visualization of significant differences between normal and bipolar individuals, particularly in the theta frequency band. This approach enhances diagnostic precision and supports personalized treatment planning, ultimately improving therapeutic outcomes for patients with bipolar disorder.