After you set $-2(x - 4)^2 + 18$ equal to 2, move the constant 18 to the other side and then divide both sides by -2 so that you have an equation involving only $(x - 4)^2$.

Once you have an equation of the form $(x - 4)^2 = \text{(number)}$, take the square root of both sides and remember to include both the positive and negative roots (the plus and minus). Then solve for $x$ by adding 4 to both sides.

You should get two expressions for $x$. Compare both of them to the answer options and see which one actually appears.

In Desmos, enter the function as y1 = -2(x - 4)^2 + 18.

On a new line, enter y2 = 2. You will see a horizontal line cutting through the parabola.

Look at the points where the parabola and the line intersect; note the x-coordinates of those intersection points (there should be two).

In Desmos, type each answer choice (for example, 4 - sqrt(2), 4 + sqrt(2), etc.) on separate lines to see their decimal values, and compare them to the x-coordinates of the intersection points. The choice whose value matches one of those x-coordinates is the solution.

We are told $q(x) = -2(x - 4)^2 + 18$ and asked for which value of $x$ we have $q(x) = 2$.

Set the expression equal to 2:

Move 18 to the right side, then divide by -2 to isolate $(x - 4)^2$.

Now solve $(x - 4)^2 = 8$ by taking square roots of both sides. Remember to include both the positive and negative roots:

Simplify $\sqrt{8}$:

So we have

Add 4 to both sides to get the two possible solutions:

From the algebra, the solutions are $x = 4 + 2\sqrt{2}$ and $x = 4 - 2\sqrt{2}$.

Among the answer options, only $4 - 2\sqrt{2}$ appears, so that is the value of $x$ that satisfies $q(x) = 2$.