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Item Black hole electromagnetic observations as tests of general relativity: quadratic gravity(Montana State University - Bozeman, College of Letters & Science, 2017) Ayzenberg, Dimitry; Chairperson, Graduate Committee: Nicolas Yunes; Kent Yagi and Nicolas Yunes were co-authors of the article, 'Linear stability analysis of dynamical quadratic gravity' in the journal 'Physical review D' which is contained within this thesis.; Nicolas Yunes was a co-author of the article, 'Slowly rotating black holes in Einstein-dilaton-Gauss-Bonnet gravity: quadratic order in spin solutions' in the journal 'Physical review D' which is contained within this thesis.; Kent Yagi and Nicolas Yunes were co-authors of the article, 'Can the slow-rotation approximation be used in electromagnetic observations of black holes?' in the journal 'Classical and quantum gravity' which is contained within this thesis.; Nicolas Yunes was a co-author of the article, 'Black hole continuum spectra as a test of general relativity: quadratic gravity' submitted to the journal 'Classical and quantum gravity' which is contained within this thesis.The recent detections of gravitational waves from merging black holes by advanced LIGO provide the first tests of General Relativity that probe the non-linear and dynamical nature of gravity. For General Relativity to be properly tested, though, many more observations are necessary. This lack of tests, coupled with several reasons General Relativity may not be the correct description of nature, motivates the study of modified theories of gravity. This dissertation presents the results of four studies on two well-motivated modified gravity theories in the class known as quadratic gravity: dynamical Chern-Simons gravity and Einstein-dilaton-Gauss-Bonnet gravity. First, I study the stability of quadratic gravity to linear perturbations. If the theory shows instabilities, black holes may not be realized in Nature, and the theory would lack physical motivation. I perform a linear stability analysis, concentrating on dynamical Chern-Simons gravity and Einstein-dilaton-Gauss-Bonnet gravity, and find that these two theories are stable to linear perturbations far from the gravitational source. Exact analytic solutions for rotating black holes in Einstein-dilaton-Gauss-Bonnet gravity are lacking, and most solutions are either numerical or approximate. I expand on previous work and find a new approximate rotating black hole solution to quadratic order in the spin angular momentum. The properties of this new solution are then studied. Many modified gravity theories lack exact solutions for rotating black holes and the approximate nature of those solutions may introduce systematic error in any attempts to constrain those theories using black hole observations. I determine the systematic error introduced by using an approximate black hole solution in General Relativity in the context of continuum spectrum and black hole shadow observations. I find that for small enough values of the spin angular momentum, the systematic error introduced is negligible compared to current sources of observational error. Finally, I study if it is possible to place better-than-current constraints on dynamical Chern-Simons gravity and Einstein-dilaton-Gauss-Bonnet gravity using black hole continuum spectrum observations. I find that while dynamical Chern-Simons gravity cannot be better constrained, with next generation telescopes it may be possible to place better constraints on Einstein-dilaton-Gauss-Bonnet gravity.