Quadratic formula for non-constant coefficients

I was driving back home and wondered if the quadratic formula still works if a, b, c are not constants? In fact, what if a, b, and c are functions of x?

0:00 Quadratic formula for nonconstant coefficients
2:30 Creating x^2+(sin(x))x-1/2*cos(x)=0
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#blackpenredpen #math #calculus #apcalculus

[socil.]

I mean what else are you supposed to think about in the car

[LegendaryBea]

The black and red pens at the back are the most underrated thing on the frame though

[willlagergaming]

The first thing i thought after seeing the thumbnail is "What!!! is this man going insane?"

[looney]

The quadratic formula is essentially just the completing the square method, which is just algebraic manipulation, so it definitely works for any equation. Whether or not completing the square is actually useful in that case, though, is a different story

[Rando]

Somebody stop him, he's getting too powerful

[taito]

" a, b, c
Qua-, dra-, tic"
😂

[boringextrovert]

I took numerical analysis back in my undergrad. One technique to solve non-standard equations, was to find an initial guess, and then somehow find a way to isolate x, and just repeat the process. Similar to Newton but more basic

[vikaaskhare]

At this point, bro is waiting for math 2 to release.

[blackpenredpen]

Newton's method (introduction & example)

[Ninja]

This gives me similar vibes to that equation you did long time ago about a quadratic equation in terms of 5.

[taterpun]

Now plug a quadratic equation into a quadratic equation

[bred]

This is really interesting as the quadratic formula can be derived just from algebraic manipulation so obviously it can work with non-constant terms. But I’ve never seen anyone use it that way and never considered doing that before either! The only condition I can think of is a=/=0 but you can just check the a=0 case before doing the quadratic. Very fascinating stuff!

[cdkw]

Hey bprp I was scrolling through your older videos and I saw that you made two promises

1. Sing the quadratic formula song!
2. Solve an easy integral in German language! :-)

Can you please do them? Maybe just a Instagram reel but please fulfill your promise!

[nicolastorres]

After getting the first solution, we can just check the original equation is even, so with that we know x_2 = - x_1.

[ianfowler]

My first reaction. It should produce a valid "value" for x for any a(x), b(x), c(x) as long as a(x)<>0. After-all it's just completeing the square. However, the solution (as you point out) may not produce anything useful or simpler. But your sin(x) example is pretty cool as you get a different equation that has the same roots - and that is definitlely very cool. The formula is sort of acting like a transformation - well in my mind anyway.

[epikherolol]

Bros taking quadratic to whole another level 💀

[XJWill]

If I was given that equation with sin(x) and cos(x)^2, I would have just used
cos(x)^2 = 1 - sin(x)^2
and then I have a quadratic equation in sin(x). Just use the quadratic formula on that. That seems more straightforward to me than using the formula on x.

[speedystrider]

You just can't get enough of abusing the quadratic equation every time you upload :)

[xTRTSCx]

Another cool example, where we would use the Lambert W function is b=3exp(x) and c=-4exp(2x), we'd get 25exp(2x) under the square root which is 5exp(x)

[cbmiami]

An interesting application would be for finding roots for equations of the form ax^3+bx+c/x, doing the trick from this video, then multiplying by x and taking the square root for a final solution.