x^2-21 = -sqrt(x^2-9)
now square both sides:
(x^2 - 21)^2 = x^2 - 9
expand the square on the left:
x^4 - 42 x^2 + 441 = x^2 - 9.
Now bring everything over to the left hand side:
x^4 - 43 x^2 + 450 =0
which is a quadratic in x^2, which may be solved using the quadratic formula
to give: x^2 = 25, or x^2 = 18.
So we have x=+/5, and x=+/-sqrt(18) are the roots, but we have to
substitute these back into the original equation because the squaring
may have introduced spurious solutions.
Doing this we find that the roots x=+/- 5 do not satisfy the original equation,
while x=sqrt(18) and x=-sqrt(18) both do.
Hence the solutions to the original equation are:
x=sqrt(18) and x=-sqrt(18).