The absence of epileptiform activity in a scalp electroencephalogram (EEG) recorded from a potential epilepsy patient can cause delays in clinical care delivery. Here we present a machine-learning-based approach to find evidence for epilepsy in scalp EEGs that do not contain any epileptiform activity, according to expert visual review (i.e., "normal" EEGs). We found that deviations in the EEG features representing brain health, such as the alpha rhythm, can indicate the potential for epilepsy and help lateralize seizure focus, even when commonly recognized epileptiform features are absent. Hence, we developed a machine-learning-based approach that utilizes alpha-rhythm-related features to classify 1) whether an EEG was recorded from an epilepsy patient, and 2) if so, the seizure-generating side of the patient's brain. We evaluated our approach using "normal" scalp EEGs of 48 patients with drug-resistant focal epilepsy and 144 healthy individuals, and a naive Bayes classifier achieved area under ROC curve (AUC) values of 0.81 and 0.72 for the two classification tasks, respectively. These findings suggest that our methodology is useful in the absence of interictal epileptiform activity and can enhance the probability of diagnosing epilepsy at the earliest possible time.