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Conceptually I am trying to understand why I end up with two radii answers of the two spheres at $\sqrt{29} \pm 1$.

I can understand how $\sqrt{29} - 1$ would give the radii of the sphere, due to it being the distance from the center of both spheres and into the radii of the radius $1$ sphere. Subtracting $1$, (the radius of the $1$ radius sphere) would yield the radius of the circle centered at the origin. But how does adding one, yield the second sphere? It just doesnt make sense as both spheres have to be the same size since they are centered at the origin and bounded by the $(3,-2,4)$ sphere.

Help.

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In one case, the two spheres are outside one another; in the other one sphere is inside the other.

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