I was stumped (but finally solved) by this seemingly simple question one of my friends challenged me, and I thought it would be good challenge for others here too :)
Q. Find for with proper steps/explanation for a 1000 sat bounty :)
There is a modern method to do it, but there's also a formula like the ol' quadratic formula for which I'll give extra sats if used :)
1,000 sats bounty
I am designating @Scroogey as the winner for this bounty with a -200 sats (for admitting to use a solver lol, although that does not give away the answer).
He showed the right steps and right equations (check second pinned comment)!
Here is another simpler walkthrough by me though (simple factorization)
or using Ferrari's method for quartic polynomials:
oh, i was gonna give it a go, but oh well haha
you still can, and I'll still give sats if you do it other than the 3 ways already done :) and this is not the last one, I'll be posting plenty....
I would have squared both sides to
and then followed https://math.stackexchange.com/questions/785/general-formula-for-solving-quartic-degree-4-equations/1219804#1219804 with a=0, b=-10, c=-1, d=20. But the quadratic equation
has no solution, indicating that I made a mistake, and I grew tired :-)
well that formula they showed is actually derived from the original quartic formula after depressing it and going case wise depending on the monic polynomial which is not the case for this one, this one is a quasi-symmetric variant, so it's not likely to give the answer :)
Good try tho! Most people won't reach till here...
Oh, this is a better strategy: https://www.mathportal.org/formulas/algebra/solalgebric.php
With a1=0,a2=−10,a3=−1,a4=20 the cubic equation becomes
and with y1=−9 (rational root test) the quadratic equations become
The solutions are
and
How did you determine the coefficients so arbitrarily? The answer is correct though but I think that's cheating. Using factor theorem is a very bad way to solve this kind of equation. Anyways 200 sats for the good effort and somehow getting the correct answer :)
I followed
from the link I posted.
that's okay, but the rational root test link shows you solved the equation using a solver which is not okay, had you not posted the link and only written "rational root test", I might not have figured it out, but since you did, it puts a wrong impression
it's okay to use a solver once you understand the method, but citing is not always necessary! (another lesson I learnt the hard way)
but another 200 sats for being honest and using formulas the correct way!
Since squaring both sides introduced false solutions, we have to check the four solutions above by inserting them into the original equation, which shows only two of them a true solutions:
and
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by "like" I mean "having a similar name to" not just the same
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