Using Substitutions to Solve Equations (Precalculus - College Algebra 23)

[@EvilMAiq]

I love that moment when it all comes together and I suddenly understand something I didn't just a moment ago. Thanks again, prof.

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you are the man professor. You are not just like a superhero, You are THE ultimate superhero. Sir I will be your patreon as soon as I realize my dreams of becoming an expert in mathematics/mathematical physics and start earning my money I promise you I am sincere about this. For now to support you wholeheartedly, I almost never skip the ads on your videos !( those which I skipped I am sorry for. ads sometimes interrupt your great lectures. I have an deep interest and excitement for mathematics haha ) Anyway I am taking notes and focusing on your videos and my books. I hope we meet someday Greetings from Turkey ! :)

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Just posting because I got a 100 on my Calc 3 final because of Professor Leonard! I’m kinda sad that calculus is over for me and he won’t be here next semester to help me through linear :( But this man helped me get through Calc 2/3 and turned me into someone who really loves math from someone who was terrified of it

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Thank you for sharing

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glad to see you safe and healthy amid the coronavirus pandemic. Thanks for taking the time to teach us mathematics.

[@amreshyadav2758]

one of the best math educator, i had ever seen. greetings from India, cheer's.

[@mattdodgers9533]

Substitution always confused me until now. Thanks so much.

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Not even once did any of my math teachers thought me this. Thank you for this! It’s very interesting!

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the superhero of mathematics! Great job as always we love you

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So, how's life now?

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[@georgesadler7830]

Professor Leonard, thank you for an outstanding lecture/video on Using Substitutions to Solve Quadratic Equations. This topic is fairly easy to follow from start to finish.

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Lookin good mate.

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So happy to see a new video, Thanks

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I am preparing for my exam in calculus and this videos have been the best guide i could dream of, thank you, thank you, thank you

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9minutes 50seconds in and i made it. Now i need to lie down in a dark room. My attention span doesn't usually last that long.

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I've been waiting for that video for a while. Thank you! Are you back on track professor?

[@HappyGardenOfLife]

I just failed calc 2 and so I'm back to learning the basics. I wish I had learned this before I took calculus.

[@thangarajselvamani178]

Thank you sir.

[@gits2807]

Thanks for all your videos in this series. Have a serious doubt at 35min into this video though. All values under an even root sign have to themselves be positive and they can yield positive or negative answers, so why would solving √x = -5 as x=(-5)^2. I understand that the value of x=25 doesn't satisfy the function like x=16 does. But I dont get the explanation you have given for x=25 being invalid. Could you please respond

[@Anthony-ui8cv]

√x, it usually refers to the principal square root, and the value is considered positive. In other words, √x is the non-negative value whose square equals x. For example, if you see √16, it typically represents 4 because 4^2 =16. While 16 also has a negative square root, which is −4 because (−4)^2 = 16, the convention is to take the positive value when we write √16. On the other hand, when you solve an equation like x^2 = a, where a is a non-negative number, the solutions for x can indeed be both positive and negative. In this context, you represent it as: x = ±√a. Hope it helps!

[@gits2807]

Thanks for the detailed explanation.. definitely helped👍

[@rosepierce9382]

Can someone tell me where I can find some notes for these lectures (Precalculus)?

[@jamesstramer5186]

It's been quite a while! Hope that you and your family are healthy and safe😄

[@99MrJ99]

awesome, thank you for explaining this.

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and He is back!

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Thank you

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[@PhillipRhodes]

Not Professor Leonard, but having gone through all of those videos, I would say that his "TTP Math" series is approximately equal to "Algebra 1" and the "Intermediate Algebra" series is approximately "Algebra 2".

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I can't find any video on inverse function of several variables ..and how the jacobian theorem comes ..about

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hello, i am so glad to see you

[@kellyhighmore2982]

We can't put a negative number under the square root but it's fine to take the square root of a number and get a negative number because if you take the square root of 25 which is x you're gonna get two solutions because -5×-5=25 & 5×5=25 so i don't understand why at x=25 we have no solution

[@0willow0]

I'm still learning, but I think it's because if you went back to plug in x=25 to check your work, you original equation would be contradictory. We ultimately need the middle term to ONLY give us -5 for it to be valid, but of course, the sqrt of 25 is also +5. Our apparent solution of x=25 causes our original equation to give us contradictory answers of 0 and 10, depending on which sqrt root of 25 the middle term equals. In short, there is no number where upon square rooting it we ONLY get -5, including 25. I think, ultimately, this is telling us that it's a double rooted parabola, because the second x=25 that the *substituted version gave us*, doesn't apply to the original. I think this type of weirdness only occurred because we used substitution. Edit: I just entered the original into Desmos, and the equation actually gives a straight line! Which kinda makes sense considering this is a power 1 equation. It seems when using substitution that we are using the substituted equation merely as a tool to find out specifically where x intercepts would be on the original (x=16), but because the original doesn't even have to be a parabola, some of the information the substitution method might tell us doesn't apply. It seems the substituted equation isn't necessarily that similar or analogous to the original beyond, in this case, one x-intercept in common.

[@cognus3372]

With the second example, you guide us to finding the imaginary solutions as well as the real solutions. In the example with x + sqrt(x) - 20, you warn us that it's a false solution. I understand that it's important to remember that the equality has to be possible with real numbers to be an intercept, but there still is a solution, though imaginary, right?

[@KhadijaAlshemeili]

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[@floortrader60]

for the 1st problem, I cam up with x = -1/3 and x = -7/3, I plugged them in the original problem and they both worked. I may have made a mistake

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Hope everyone is safe

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Our hero is back in black........

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Welcome back!

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OMG please do linear algrebra, PRETTY PLEASE!

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Eigenvalues arrgh

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Thank you sir your videos are awesome and they helped me a lot I have a request to you that how may I contact to you I have some doubts and where can I send those !!??

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Where have u been prof?

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✍️

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Sir why don't you post videos on differential calculus

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Done 🙂

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I have a problem understanding math, hey, where do we use all the equations💔

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Almost everything you can think of, from nature to technology has a mathematical blueprint guiding its functionalities or operates on mathematical logic at its most fundamental level..

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I’m so sorry to post it here. What is the difference of the word "trigonometry" and "trigonometric"?

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First is a noun; second is an adjective. "Sine is a function commonly used in _trigonometry_ ". "Sine is a _trigonometric_ function".

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U substitution?

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Right side was quiet this week XD

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liked by teacher

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I wasnt planning on finishing the video, but I liked it. I came to ask if you had a mustache in your calculus videos. I cannot remember if you did

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The sexy version of Stephen Colbert.

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I know I wouldn't!!

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dude, no, there are kids watching these videos...

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