Question 919·Hard·Math

Type both equations into Desmos and click each crossing. Desmos shows approximately $(-1,3)$, $(-0.618,2.618)$, and $(1.618,0.382)$. An intersection is on both graphs, so each first coordinate is an $x$-value that makes their $y$-values equal. The graph shows which point was given, but its rounded labels aren't needed for the exact sum.

At each crossing, the two expressions for $y$ are equal, so set them equal:

Subtract $2-x$ from both sides to put the equation in standard form, with $0$ on one side:

Call the other two $x$-coordinates $r$ and $s$. A root makes the cubic $0$, so the roots $-1$, $r$, and $s$ give it the factors $x+1$, $x-r$, and $x-s$. Multiplying shows the $x^2$ coefficient:

The equation $x^3-2x-1=0$ has no $x^2$ term, so that coefficient is $0$. The missing $x^2$ term means all three roots add to $0$:

The $-1$ is the given point's $x$-coordinate, not one of the two you need. Subtract $-1$ from the total:

Simplify the double negative:

So the other two intersection points' $x$-coordinates sum to $1$. Grid in 1.