Question 22·Medium·Advanced Math

An $x$-intercept is where the graph meets the horizontal axis, so its output is $0$. Here $y=f(x)$, so set $f(x)=0$ to find an $x$-intercept:

The word all means you need every $x$-value that works.

Type $f(x)=-2(x-3)^2+10$ in Desmos. Click the graph, then click both marked points on the $x$-axis. Desmos shows approximately $(0.764,0)$ and $(5.236,0)$. Their first coordinates are the $x$-values to match.

Add $3-\sqrt5$ and $3+\sqrt5$ on separate Desmos lines. They print approximately $0.764$ and $5.236$, matching both crossings. So the $x$-intercepts have $x$-values $3-\sqrt5$ and $3+\sqrt5$. Choice C.

No Desmos needed. The single square in the given function lets you find exact roots by hand. From $0=-2(x-3)^2+10$, subtract $10$ from both sides:

Divide both sides by the coefficient $-2$:

Take the square root of both sides. A positive and a negative number can have the same square, so keep both signs:

Add $3$ to both sides:

That gives both $x$-intercepts: $3-\sqrt5$ and $3+\sqrt5$. Choice C.