Quadratic formula for non-constant coefficients @blackpenredpen

blackpenredpen | Quadratic formula for non-constant coefficients @blackpenredpen | Uploaded 3 months ago | Updated 5 minutes ago
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?
Newton's method (introduction & example) youtu.be/iVOsU4tnouk

0:00 Quadratic formula for nonconstant coefficients
2:30 Creating x^2+(sin(x))x-1/2*cos(x)=0
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$integral of 1/(x^2+1) but you didnt learn it this way in calculus 2 Learn more complex numbers by visiting Brilliant 👉 https://brilliant.org/blackpenredpen/ (20% off with this link!) 🛍 Eulers Identity e^(iπ)+1=0 t-shirt: https://amzn.to/427Seae When you want to use complex numbers to integrate 1/(x^2+1)! We didnt use partial fraction decomposition with complex numbers to integrate 1/(x^2+1) in calculus 2. We usually use a trigonometric substitution to integrate 1/(1+x^2), i.e. the tangent substitution. But here the key is to simplify the complex logarithms and get their principal branch. Be sure to look at the polar form of the complex numbers. Then we can integrate 1/(1+x^2) and get arctan(x), just as expected! This is definitely a great challenge question for any calculus student! ***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do*** AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist Robert Sundling Stuart Wurtman Nick S William OCorrigan Ron Jensen Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L 💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here https://www.patreon.com/blackpenredpen$

use trigonometric substitution, not power rule

$100 Inevitable Algebra Problems for Calculus 100 Inevitable Algebra Problems for Calculus! Designed for AP calculus or college calculus students, this tutorial covers a wide range of topics including factoring trinomials, polynomial long division, synthetic division, logarithmic functions, exponential functions, and more. Youll also learn techniques for simplifying complex expressions and solving partial fraction decompositions. Whether youre looking to improve your math skills or ace your calculus exams, this video has everything you need. #APCalculus #CollegeCalculus #AlgebraProblems #math #blackpenredpen the file: https://www.blackpenredpen.com/_files/ugd/287ba5_88ffffbd539d419a83482923d2352fbf.pdf with answers: remind me BIG THANKS TO Alex Zyryanov for the timestamps! 0:00 the extreme algebra workout for calculus! Q1 to Q10 factor & cancel 0:46 - Q1 2:05 - Q2 4:33 - Q3 6:33 - Q4 9:39 - Q5 11:52 - Q6 15:12 - Q7 23:51 - Q8 25:40 - Q9 27:24 - Q10 Q11 to Q20 expand & cancel 30:21 - 11 39:06 - 12 43:29 - 13 45:40 - 14 49:29 - 15 53:19 - 16 55:48 - 17 58:52 - 18 1:01:12 - 19 1:05:00 - 20 Q21 to Q30 using the conjugate for square root expressions 1:08:15 - 21 1:11:19 - 22 1:13:27 - 23 1:17:00 - 24 1:19:35 - 25 1:22:05 - 26 1:25:22 - 27 1:26:41 - 28 1:30:19 - 29 1:33:31 - 30 Q31 to Q40 complex fractions 1:36:13 - 31 1:39:32 - 32 1:42:30 - 33 1:44:53 - 34 1:47:32 - 35 1:49:56 - 36 1:51:41 - 37 1:54:34 - 38 1:56:36 - 39 1:59:34 - 40 Q41 to Q50 LCD & combine 2:03:16 - 41 2:05:04 - 42 2:07:43 - 43 2:10:27 - 44 2:13:03 - 45 2:17:05 - 46 2:19:52 - 47 2:22:37 - 48 2:25:56 - 49 2:29:47 - 50 Q51 to Q60 exponential & logarithm 2:34:25 - 51 2:38:06 - 52 2:42:15 - 53 2:44:48 - 54 2:46:13 - 55 2:47:53 - 56 2:49:18 - 57 2:50:29 - 58 2:53:30 - 59 2:55:40 - 60 Q61 to Q70 write in terms of x^p 2:57:23 - 61 2:59:04 - 62 2:59:58 - 63 3:01:00 - 64 3:02:58 - 65 3:04:40 - 66 3:06:00 - 67 3:08:00 - 68 3:09:09 - 69 3:12:22 - 70 Q71 to Q80 expressions from the quotient rule of derivative 3:14:09 - 71 3:16:25 - 72 3:19:29 - 73 3:22:19 - 74 3:24:10 - 75 3:26:16 - 76 3:29:50 - 77 3:32:58 - 78 3:36:29 - 79 3:39:15 - 80 Q81 to Q90 expressions from implicit differentiation (solving for dy/dx) 3:43:40 - 81 3:47:55 - 82 3:49:53 - 83 3:52:28 - 84 3:55:13 - 85 3:59:08 - 86 4:02:44 - 87 4:09:32 - 88 4:12:13 - 89 4:15:28 - 90 Q91 to Q100 solving for the unknowns from partial fractions decomposition 4:18:22 - 91 4:26:31 - 92 4:33:42 - 93 4:38:42 - 94 4:43:56 - 95 4:46:58 - 96 4:48:33 - 97 4:53:09 - 98 4:59: 46 - 99 5:04: 20 - 100 5:14:33 for the people who just want to see the end : ) ***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do*** AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist Robert Sundling Stuart Wurtman Nick S William OCorrigan Ron Jensen Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor. Jasmine Soni L wan na Marcelo Silva Evangeline McDonald Joe Taxpayer 💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here https://www.patreon.com/blackpenredpen$