The college board has come out with the AP PreCalculus course. I haven't made my mind up (the details seem sketchy so far)
https://apcentral.collegeboard.org/courses/ap-precalculus/about-ap-precalculus
So we're ready to subsitute
\(p = \dfrac{p + q}{2} + \dfrac{p -q}{2} = \dfrac{-b}{2a} + \dfrac{\sqrt{ \left(\dfrac{-b}{a}\right)^2 - 4 \cdot \dfrac{c}{a}}}{2} \)
That simplifies to:
\( p = \dfrac{-b}{2a} + \dfrac{\sqrt{b^2 - 4ac}}{2a} \)
And the same goes for q.
\( a^2 + bx + c = 0 \)
From Vieta for the roots p,q: \(p +q = -\dfrac{b}{a} \text{ and } p\cdot q = \dfrac{c}{a} \)
Next each root root is symmetrically around the axis of symmetry through the vertex. Algebraically this is also easily verifiable.
\(p = \dfrac{p + q}{2} + \dfrac{p -q}{2} \)
We now just need an expression for p - q when we only have one for p + q
But
\( (p+q)^2 - 4pq = (p-q)^2 \)
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Also random notes
1. There doesn't seem to be way to find out how many times a toot has been seen (Even locally on this instance)
2. I don't understand how the forwarder bots on birdsite.wilde.cloud etal are setup. Can you get more accounts added?
3. Its a bit limiting not being able to search for terms to find things. No looking for other people who have discussed group theory unless they hashtagged it.
1st in the series. I've been collecting these for multiple years on https://blog.mathoffthegrid.com/p/collected-problems-2.html (and several followup pages)
Ah and I almost forgot I have a blog at blog.mathoffthegrid where I recently hosted one of the carnivals of Math but normally post about whatever mathy things I'm thinking about
One of the things I'm still looking for here is if anyone posts fun problems ala Catriona Agg etc,
A quick #introduction: I've been leading math circles mostly for middle school students the last five years,. So I'm always lookout for interesting activities or ideas to try out.
I host the local #mathsjam for Seattle. Recently we've been doing things virtually with the folks in Spokane and I've enjoyed getting to know the folks on that side of the state. Although I still very much miss in person get togethers and hope to do them again this summer.
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Also there seem to be some sort of bots now that can retoot things from twitter which I should look at
Systems programmer, parent, volunteer Math Circle leader and general math enthusiast. Team puzzles and #geometry Honorary Lord Semichord
@benjamin_leis back on twitter
Located in Seattle, WA
blog.mathoffthegrid.com