If the distance between the two points 𝑎, zero and negative 𝑎 plus one, zero is nine, find all possible values of 𝑎.
So if we are finding the distance between two points, 𝑥 one, 𝑦 one would be the first point and 𝑥 two, 𝑦 two would be the second point. Then we take the square root and do 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared and again you take the square root of all of that. So we are told one of the points is 𝑎, zero and the other point is negative 𝑎 plus one, zero and our distance is equal to nine.
So first, we have 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared and our distance between these is equal to nine and we’re trying to find all the values of 𝑎. So let’s evaluate everything. We can combine the negative 𝑎s and we have plus one and we square this. And zero minus zero is zero, so we have zero squared again equals nine. So before we move on, zero squared is just zero, so it disappears. So we have the square root of negative two 𝑎 plus one squared equal to nine. And the square root of something squared makes the square and the square root just disappear.
However, we need to remember the fact that whatever 𝑎 is, since we are squaring it, it would be okay to plug in any answer that is negative. And it could also be positive because if you square a negative or you square a positive, it will equal a positive. So we want to take negative two 𝑎 plus one and set it equal to positive nine and negative nine. Now, if that doesn’t make enough sense, we can look at this in a different way. So let’s go back to this step.
So let’s say we didn’t take the square root and cancel with the square; let’s say instead we squared both sides to get rid of the square root. So then we would have negative two 𝑎 plus one squared cause the square root’s gone equal to nine squared, which is 81. Now to solve for 𝑎, our next step would be to square root both sides. So the square and the square would cancel. Now the square root of 81 is positive nine and it’s negative nine, so we’re exactly where we were. That’s just another route to go. So that means we need to take negative two 𝑎 plus one to be set equal to positive nine and negative two 𝑎 plus one should be set equal to negative nine.
So our first step would be to subtract one from both sides. So we have negative two 𝑎 equals positive eight and negative two 𝑎 equals negative 10. And now we divide both sides by negative two. So we get 𝑎 equals negative four and we also get that 𝑎 equals five. Now when we write our final answer, we should write 𝑎 equals negative four or 𝑎 equals negative five. You don’t wanna plug in both of them at the same time, so you write or.