$x^2 + 4y^2 + 2x - 4y + 2 = 0$
$x^2 + 4y^2 + 2x - 4y + 1 + 1 = 0$
$(x^2 + 2x + 1) + (4y^2 - 4y + 1) = 0$
$(x + 1)^2 + (2y - 1)^2 = 0$
x + 1 = 0
x = −1
2y − 1 = 0
2y = 1
$y = \frac{1}{2}$

$9x^2 + y^2 - 12x + 8y + 21 = 0$
$9x^2 + y^2 - 12x + 8y + 16 + 4 + 1 = 0$
$(9x^2 - 12x + 4) + (y^2 + 8y + 16) + 1 = 0$
$(3x^2 - 2)^2 + (y + 4)^2 + 1 = 0$
$(3x^2 - 2)^2 + (y + 4)^2 + 1 ≠ 0$, так как:
$(3x^2 - 2)^2 ⩾ 0$;
$(y + 4)^2 ⩾ 0$, следовательно:
$(3x^2 - 2)^2 + (y + 4)^2 + 1 > 0$.