Here it is: ruclips.net/video/cUzklzVXJwo/видео.html It's about the solution of cubic equations, which was an even more difficult find than the quadratic formula.
That formula is quite simple: consider substitution x=y−b/(2a), so you get “depressed quadratic” ay²−(b²−4ac)/(4a), and the rest is trivial. Moreover, substitution in form of x=y−aₙ₋₁/(naₙ) eliminates term with n−1'th power in aₙxⁿ+aₙ₋₁xⁿ⁻¹+aₙ₋₂xⁿ⁻²+…+a₁x+a₀ simply because of binomial expansion.
formula wise they began with the standard quadratic equation which is this ax^2 + bx + c = 0 and solved for x. so first solve for x by subtracting c from both sides. second divide by a. [ now you have x^2 + bx/a = -c/a] third complete the square by adding (b/2a)^2 to both sides. [now you have x^2 + bx/a + (b/2a)^2 = -c/a + b^2/4a^2]. fourth now the left side can be written as (x+b/2a)^2. fifth the right side uses LCD to become (-4ac + b^2)/4a^2. sixth take the square root of both sides and get x + b/2a on the left and the square root of (b^2 - 4ac)/2a on the right. Lastly subtract b/2a from both sides and combine terms on the right to get x = - b +- square root of (b^2 - 4ac)/2a
could you make a longer video about how you approach these kinds of problems from a fresh start? I like how you explain the process but sometimes I wonder how you figured out a lot of it. love ur vids btw
it's mostly experience and a shit load of honest homework solving. After you've seen several hundreds of geometry problems then you just see the patterns and possible ways for solutions
My guess is try and look for triangles, and remember that two circles with a tangent point can have their radiuses added to form a straight line through both
Unrelated to this video, but the page of your website(love that site, btw) for the “Trapezoid- Middegment (or Median)” has a typo in the Trapezoid Midsegment Theorem description. You accidentally have “lengtsh” instead of “lengths” Love all the math content and the website!!
In the quadratic formula shown in the video, how could we interpret the negative result? I know there's no negative lenght, but the solution with negative number has to have a meaning.
Only that it's a hypothetical mathematical answer that, when you evaluate it in the practical conditions to be used, can't apply. Simple analogy, consider the statement "The parent who gave birth to you is sick." Imagine you were adopted by a married (straight) couple. The word "parent" then, has multiple possible definitions: your birth father, your birth mother, your adoptive mother, or your adoptive father. Theoretically, any of them could be the person who is sick. However, the additional condition "gave birth to you" restricts it to only one answer, your birth mother. While the others are theoretically answers, practically they are not. The incorrect answers don't have any meaning, they're just incorrect given the circumstances.
can you show how to trisect an angle exactly. i'm not interested in the classical problem, we know that is not possible. marked ruler or whatever, just exact trisection
This gets asked in almost every video with tangent circles. For any circles that are tangent (either internally or externally), the centres of each circle and the tangent point are colinear. It is a basic circle theorem. Funny how no one ever askes if they should be showing that Pythagoras is valid before using it.
@@Grizzly01-vr4pn no need to be snarky at the end dude. It's a valid question, especially when you watch the first video and notice that the radius Andy made didn't form a straight line.
@@troybaxter Snarky or a valid observation? Also, I _have_ watched the first video, and I'm not sure what you're talking about. The two radii (lengths = 5 and 8) clearly form a straight line, which is then taken as a hypotenuse of length = 13. 🤷♂
x^2 + x^2 - 10x + 25 = 169 2 x^2 - 144 = 10 x x^2 - 72 = 5 x (x^2)^2 - 2 * 72 * x^2 + 72^2 = 25 x^2 y = x^2, find y y^2 -144 y + 72^2 = 25 y y^2 - 169 y + 72^2 = 0 y = (169 +/- sqrt(169^2 - 4 * 1 * 72^2)) / (2 * 1) y = (169 +/- sqrt(169^2 - 144^2)) / 2 y = (169 +/- sqrt((169 + 144)(169 - 144)) / 2 y = (169 +/- sqrt(313 * 25) / 2 y = (169 + 5 * sqrt(313)) / 2 Negative value in +/- is dropped because side would be less than 8 if we chose it as solution, making it extraneous.
The quadratic formula always blows my mind - not that it works but like… how tf did someone come up with that shit??
They always go through the proof in class of how to get the equation.
And also, there's a Veritasium video about a very similar topic. One sec.
Here it is:
ruclips.net/video/cUzklzVXJwo/видео.html
It's about the solution of cubic equations, which was an even more difficult find than the quadratic formula.
That formula is quite simple: consider substitution x=y−b/(2a), so you get “depressed quadratic” ay²−(b²−4ac)/(4a), and the rest is trivial.
Moreover, substitution in form of x=y−aₙ₋₁/(naₙ) eliminates term with n−1'th power in aₙxⁿ+aₙ₋₁xⁿ⁻¹+aₙ₋₂xⁿ⁻²+…+a₁x+a₀ simply because of binomial expansion.
formula wise they began with the standard quadratic equation which is this ax^2 + bx + c = 0 and solved for x. so first solve for x by subtracting c from both sides. second divide by a. [ now you have x^2 + bx/a = -c/a] third complete the square by adding (b/2a)^2 to both sides. [now you have x^2 + bx/a + (b/2a)^2 = -c/a + b^2/4a^2]. fourth now the left side can be written as (x+b/2a)^2. fifth the right side uses LCD to become (-4ac + b^2)/4a^2. sixth take the square root of both sides and get x + b/2a on the left and the square root of (b^2 - 4ac)/2a on the right. Lastly subtract b/2a from both sides and combine terms on the right to get x = - b +- square root of (b^2 - 4ac)/2a
Man didn't miss the opportunity to say nice. Nice.
could you make a longer video about how you approach these kinds of problems from a fresh start? I like how you explain the process but sometimes I wonder how you figured out a lot of it. love ur vids btw
Yes please! I need to learn his way of thinking
it's mostly experience and a shit load of honest homework solving. After you've seen several hundreds of geometry problems then you just see the patterns and possible ways for solutions
My guess is try and look for triangles, and remember that two circles with a tangent point can have their radiuses added to form a straight line through both
Keeping my math fresh over summer lol
You are doing a great job, I support you, keep it up. Thank you very much for teaching us. ❤❤❤
I gather these problems for about a week and binge solve them and mostly fail 😂
Look at the window
Way too complicated.
S = Side length
R = 8
r = 10/2 = 5
S² + (S - r)² = (R + r)²
S² + (S - 5)² = (8 + 5)²
Yep, I also solved it this way
This one was much easier than finding the length of the tangent
Please calculate the area of the white part in the next part also, while in the process finding area of the part of circles
This looks important. Let's put a box around it. 🎁
Please calculate the area of the white part in the next part also, while in the process finding area of the part of circles.
AofCircle=pi*r^2 AoftheSquare=128.72951
Aof5Circle=(pi*5^2)/2=39.3
Aof8Circle=(pi*8^2)/4=50.3
AoftheSquare-Aof5Circle-Aof8Circle=AofWhite
= 128.7-39.3-50.3
=39.1
Unrelated to this video, but the page of your website(love that site, btw) for the “Trapezoid- Middegment (or Median)” has a typo in the Trapezoid Midsegment Theorem description. You accidentally have “lengtsh” instead of “lengths” Love all the math content and the website!!
Thank you, I fixed it.
Please find the area of the white part in the next part also, while in the process finding area of the part of circles.
Tomorrow I have a calculus exam so I will use your channel to revise
How do we know that the line from the bottom right corner to the middle of the semicircle passes thru the tangent point?
How exciting!
In the quadratic formula shown in the video, how could we interpret the negative result? I know there's no negative lenght, but the solution with negative number has to have a meaning.
Only that it's a hypothetical mathematical answer that, when you evaluate it in the practical conditions to be used, can't apply. Simple analogy, consider the statement "The parent who gave birth to you is sick." Imagine you were adopted by a married (straight) couple. The word "parent" then, has multiple possible definitions: your birth father, your birth mother, your adoptive mother, or your adoptive father. Theoretically, any of them could be the person who is sick. However, the additional condition "gave birth to you" restricts it to only one answer, your birth mother. While the others are theoretically answers, practically they are not. The incorrect answers don't have any meaning, they're just incorrect given the circumstances.
can you show how to trisect an angle exactly. i'm not interested in the classical problem, we know that is not possible. marked ruler or whatever, just exact trisection
I solved it very easily
But the numbers are quite dirty. I solved it by my hands..
❤❤❤
I love this channel!
You're recording a ton of videos in a batch, right?
Easy Sneezy.
Please find the area of the circle.
just visually looking at it you know it will be greater than 80 units and less than 130 units. thats as far as I got.
This guy will see a photo, ask you for the age of your mother, and figure out the blood type of your future girlfriend
How exciting
Nice, but don’t you have to show that the two radii form a straight line?
This gets asked in almost every video with tangent circles.
For any circles that are tangent (either internally or externally), the centres of each circle and the tangent point are colinear. It is a basic circle theorem.
Funny how no one ever askes if they should be showing that Pythagoras is valid before using it.
@@Grizzly01-vr4pn no need to be snarky at the end dude. It's a valid question, especially when you watch the first video and notice that the radius Andy made didn't form a straight line.
@@troybaxter Snarky or a valid observation?
Also, I _have_ watched the first video, and I'm not sure what you're talking about. The two radii (lengths = 5 and 8) clearly form a straight line, which is then taken as a hypotenuse of length = 13.
🤷♂
They're both perpendicular to the same line (which is tangent to both circles). Therefore the angle between them is 90+90=180 degrees - straight line.
x^2 + (x-5)^2 = 13^2
find x^2
x^2 + x^2 - 10x + 25 = 169
2 x^2 - 144 = 10 x
x^2 - 72 = 5 x
(x^2)^2 - 2 * 72 * x^2 + 72^2 = 25 x^2
y = x^2, find y
y^2 -144 y + 72^2 = 25 y
y^2 - 169 y + 72^2 = 0
y = (169 +/- sqrt(169^2 - 4 * 1 * 72^2)) / (2 * 1)
y = (169 +/- sqrt(169^2 - 144^2)) / 2
y = (169 +/- sqrt((169 + 144)(169 - 144)) / 2
y = (169 +/- sqrt(313 * 25) / 2
y = (169 + 5 * sqrt(313)) / 2
Negative value in +/- is dropped because side would be less than 8 if we chose it as solution, making it extraneous.
First personn
bro no one cares XD
How exciting!
Second person .
Please calculate the area of the white part in the next part also, while in the process finding area of the part of circles