What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A( -2, 0)$ and $B(0,4)$?
To find the point on the $x$-axis that is equidistant from $A$ and $B$, I need to find a point $(x,0)$ such that the distance from $(x,0)$ to $A$ is equal to the distance from $(x,0)$ to $B$. I can use the distance formula to write an equation for this condition: $\sqrt{(x+2)^2+(0-0)^2}=\sqrt{(x-0)^2+(0-4)^2}$. Squaring both sides of the equation, I get $(x+2)^2=x^2+16$. Expanding the left side and simplifying, I get $x^2+4x+4=x^2+16$, or $4x=12$. Dividing both sides by $4$, I get $x=3$. Therefore, the $x$-coordinate of the point on the $x$-axis that is equidistant from $A$ and $B$ is $3$. 