In triangle ABC, ∠ABC = 90◦ . Point M is on side AB so that AM = MB, and CM = 4 and AC = 7. Find BC

Let x = BC and y = AM = MB. By Pythagorean theorem we have \(\begin{cases}(2y)^2 + x^2 = 7^2\\y^2 + x^2 = 4^2\end{cases}\).

Subtracting the second equation from the first gives an equation solely in y^2.

Can you take it from here?