We investigate the clustering properties of ~1550 broad-line active galactic nuclei (AGNs) at <z>=0.25 detected in the ROSAT All-Sky Survey (RASS) through their measured cross-correlation function with ~46,000 Luminous Red Galaxies (LRGs) in the Sloan Digital Sky Survey. By measuring the cross-correlation of our AGN sample with a larger tracer set of LRGs, we both minimize shot noise errors due to the relatively small AGN sample size and avoid systematic errors due to the spatially varying Galactic absorption that would affect direct measurements of the auto-correlation function (ACF) of the AGN sample. The measured ACF correlation length for the total RASS-AGN sample (<L_(0.1-2.4 keV)>=1.5 x 10^(44) erg/s) is r_0=4.3^{+0.4}_{-0.5} h^(-1) Mpc and the slope gamma=1.7^{+0.1}_{-0.1}. Splitting the sample into low and high L_X samples at L_(0.5-10 keV)=10^(44) erg/s, we detect an X-ray luminosity dependence of the clustering amplitude at the ~2.5 sigma level. The low L_X sample has r_0=3.3^{+0.6}_{-0.8} h^(-1) Mpc (gamma=1.7^{+0.4}_{-0.3}), which is similar to the correlation length of blue star-forming galaxies at low redshift. The high L_X sample has r_0=5.4^{+0.7}_{-1.0} h^(-1) Mpc (gamma=1.9^{+0.2}_{-0.2}), which is consistent with the clustering of red galaxies. From the observed clustering amplitude, we infer that the typical dark matter halo (DMH) mass harboring RASS-AGN with broad optical emission lines is log (M_DMH/(h^(-1) M_SUN)) =12.6^{+0.2}_{-0.3}, 11.8^{+0.6}_{-infty}, 13.1^{+0.2}_{-0.4} for the total, low L_X, and high L_X RASS-AGN samples, respectively.
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