We establish necessary and sufficient conditions for the solvability to the matrix equation and present an expression of the general solution to (1) when it is solvable. As applications, we discuss the consistence of the matrix equation where * means conjugate transpose, and provide an explicit expression of the general solution to (2). We also study the extremal ranks of X ₃ and X ₄ and extremal inertias of and in (1). In addition, we obtain necessary and sufficient conditions for the classical matrix equation to have Re-nonnegative definite, Re-nonpositive definite, Re-positive definite and Re-negative definite solutions. The findings of this article extend related known results.