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I'm studying for an upcoming multi-variable Calculus exam and I'm struggling to solve the following problem:

Find an equation for the surface consisting of all points that are equidistant from the point $(-1, 0, 0)$ and the plane $x = 1$. Identify the surface.

I've found a few solutions online (including one on this stackexchange) but they all use a distance formula I don't recognize. I am familiar with the formula:

$D\:=\:\frac{\left|a\left(x_1\right)+b\left(y_1\right)+c\left(z_1\right)+d\right|}{\sqrt{a^2+b^2+c^2}}$

The formula I see is:

$D\:=\:\sqrt{\left(x^2+1\right)^2+y^2+z^2}$

Is this essentially the formula computing the distance of the line between $x = 1$ and the point $(1,0,0)$? Where does this formula come from? It reminds me for the formula for the length/magnitude of a vector, and I can see that $(x^2-1)^2$ is linked to x being -1 in the point $(-1,0,0)$, but beyond that I'm not sure why we are using that instead of the first distance formula I posted above.

EDIT: So after some digging around I realized that this is just the formula used to compute distance between two points. However I'm not sure why we are computing the distance between $(x,y,z)$ and $(-1,0,0)$ as well as the distance between $(x,y,z)$ and $(1,y,z)$. For some reasoning I'm having a devil of a time understanding what is happening.

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    The distance between the plane $P=\{x=1\}$ and the point $A=(1,0,0)$ is irrelevant. Instead, one must compute the distance $d(M,P)$ between some point $M=(x,y,z)$ and the plane $P$ and ask that $d(M,P)=MA$, or, better yet, that $d(M,P)^2=MA^2$. Which step is a problem? – 2017-02-13
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    For insight, look at the corresponding two-dimensional situation, where you want to understand the curve consisting of all points that are equidistant from the point $(-1,0)$ and the line $x=1$. I’m sure you’ve already seen the answer to this question. – 2017-02-13
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    To add to @Lubin’s comment, all that’s added in moving to three dimensions from the two-dimensionional case is radial symmetry. – 2017-02-13
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    And to add to @amd’s comment, make sure you draw a picture. This is most important. – 2017-02-13

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