SAT Question 0980fcdd: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

Choice A is correct. Solutions to the given system of equations are ordered pairs $(x,y)$ that satisfy both equations in the system. Adding the left-hand and right-hand sides of the equations in the system yields $x^2+y=6x+(-6x)+y+36$, or $x^2+y=y+36$. Subtracting $y$ from both sides of this equation yields $x^2=36$. Taking the square root of both sides of this equation yields $x=6$ and $x=-6$. Therefore, there are two solutions to this system of equations, one with an $x$-coordinate of $6$ and the other with an $x$-coordinate of $-6$. Substituting $6$ for $x$ in the second equation yields $y=-6\times 6+36$, or $y=0$; therefore, one solution is the ordered pair $(6,0)$. Similarly, substituting $-6$ for $x$ in the second equation yields $y=-6\times (-6)+36$, or $y=72$; therefore, the other solution is the ordered pair $(-6,72)$. It follows then that if the ordered pair $(x,y)$ is a solution to the system, then possible values of $xy$ are $6\times 0=0$ and $-6\times 72=-432$. Only $0$ is among the given choices.