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I've never encountered such a question before; it was included in as a sub-question on a different topic.

I think I'd understand how I'd get 6t^2 and 125 (square then sum the individual numbers in each coordinate) but no idea how to get -24t. What is the correct way of approaching/doing this question?

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The distance between the points $A$ and $P$ is given by the Pythagorean formula

$$\sqrt{(8-t)^2+(-6-t)^2+(5-2t)^2}\;,$$

so the square of the distance is

$$(8-t)^2+(-6-t)^2+(5-2t)^2\;.$$

What do you get when you simplify this polynomial?

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    Thank you. I had originally attempted to try something vaguely similar to this (squared each coordinate of P before trying to find the distance).2012-12-12
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    @Wk_of_Angmar: You’re welcome.2012-12-12