Question #ae3bc
1 Answer
Compare with a general second degree polynomial (a quadratic equation), then look at the discriminant of the quadratic formula and find the condition that
Explanation:
Ah, that happens to be a quadratic equation! Compare:
With what we have:
It seems to be a special case where
- Treat it like a quadratic equation and directly use the quadratic formula etc.
- Play around with it like it was any other polynomial
Hmm, quite a difficult decision... I'll do both, firstly doing the first option for readers with a limited amount of time, then (also) the second option for leisure readers with plenty of time.
Well, we need to solve for
For any
Our polynomial is a special case of this where
Currently, we are asked about which conditions where the "zeros" of this polynomial exist. That means what the
However, we need to make sure we aren't getting imaginary numbers so that it appears in the graph, since it should be for real numbers. How do we do that? Well, let's look inside the metaphorical possible breeding spot for imaginary numbers: the radical!
We need to make sure that whatever number that shows up in there isn't negative!
So, that's the condition! You could rearrange, if needed:
Well, we have
And we first want to solve for
Now, there are two
Hmm... let's try to match it with what we have! We could turn
Rearrange this a bit:
Then... add
Now we could match it! We could have
So let's substitute this back:
Nice! We only have one
Subtract
Finally, add back
Ah, that gets us back to where we started! Good indicator that we're in the right direction, but we're not really getting anywhere, so... how about if we don't expand the parenthesis containing
Now, to really "take the
And add
Eh, maybe there's something we can do with what's inside the radical... Maybe make
What I just did here, is multiply
Then we can "split" the square root:
Whew! Now we focus on the contents of the radical:
Which shouldn't be a negtive number, otherwise we'd have imaginary numbers, and such
Rearranging, by adding
Ah, so that's our condition!
By the way, what I just did was a "special case" for a general method used to derive the quadratic formula, which can be seen here: