# Thread: what is happening here?

1. Originally Posted by allegansveritatem
The second sentence of the above is certainly NOT true, right? I mean, when is it ever true?
I strode past that part--the part where the negative number appeared on one side and a lonely radical on the other--like a giant racing to the finish line. When I say "like a giant" you should see in imagination a giant rhinoceros in mid charge.
$a + b = c \implies a = b + c \text { if and only if } b = 0.$

It is why I kept saying generally; there is that one special case.

2. Originally Posted by allegansveritatem
Well, I checked in the sense that it was easy to see that 5 was the obvious and only answer. But I thought that when an equation has only one answer, like this one, there can be no extraneous root to check, no?
My comment about not checking was to Denis, not you. I have assumed all along, from what you have said, that you not only knew what the book said, but had seen that 5 actually works.

But, NO! You never know whether an equation has an answer at all. You can't say "it has only one [possible?] answer, so it can't be extraneous"; there might be no answer. And if the book shows only one answer, it might be wrong. You have to check even then.

If your problem had been a subtraction rather than an addition on the left, then there would be no solution; you would have found the 5, but it would be extraneous.

3. Originally Posted by allegansveritatem
But I thought that when an equation has only one answer, like this one, there can be no extraneous root to check, no?
That is not true. Can you think of an example?

How about -$\sqrt{x+1}$ = $\sqrt{x^2 - 5x + 10}$?

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