Substitute $y = -6$ into $x^2 + y^2 = 100$ and simplify. You should end up with an equation involving only $x^2$.

When you solve for $x$ from an equation like $x^2 = 64$, remember that there are two solutions, one positive and one negative. Then see which of those values appears in the answer choices.

In Desmos, enter x^2 + y^2 = 100 to graph the circle centered at the origin with radius 10.

On a new line, enter y = -6 to graph the horizontal line that intersects the circle.

Click on one of the intersection points of the line and the circle. Desmos will display the coordinates of the point; note the $x$-values of these intersection points and choose the matching value from the answer choices.

The circle is given by

and the line is given by $y = -6$.

At the intersection point, both equations are true at the same time. So replace $y$ in the circle equation with $-6$:

Compute $(-6)^2$:

So the equation becomes

Subtract 36 from both sides:

Now you know that $x^2 = 64$.

To solve $x^2 = 64$, take the square root of both sides. Remember that both the positive and negative roots are possible:

The question asks for one possible value of $x$ and gives the answer choices $-6$, $6$, $10$, and $-8$.

From $x = \pm 8$, the value that appears in the list is $-8$, so the correct answer is $-8$.