Lot of confusion in the comments about the p and q part. Quick written explanation for anyone not sure what happened in that step, it can be reduced from a quartic to a simple quadratic with a substitution of a=x². Solving a² - 78a + 1296 = 0 using the quadratic formula gives the roots.
It might be worth noting that before we dive into the algebra, we can infer some properties of the solutions. We are trying to solve two simultaneous equations whose solutions correspond to the intersection of a circle of radius SQRT(78) and a hyperbola with vertices at (6,6) and (-6,-6); we know that any solutions will be mirrored in the first and third quadrants of the cartesian plane. We can also infer that since x and y are interchangeable in the equations, they must be so in the solutions (i.e., you should be able to swap the x & y values of a solution to get another solution - another way of saying the solutions are reflected in the line y = x). Finally, we can infer there must be 4 real solutions (two in each of quadrant 1 and quadrant 3), rather than zero or 2 (one in each quadrant), by noting that the circle’s radius is greater than SQRT(2*36), so it must intersect the hyperbola at two points in both quadrants (draw yourself a diagram of the two equations in the cartesian plane to see this). If the radius was less than SQRT(72), there would be no intersections (i.e., zero real solutions), and if it was exactly SQRT(72) (i.e., x^2 + y^2 = 72), then there would be exactly one solution in quadrant 1 and one in quadrant 3 (i.e., at the hyperbola’s vertices). Of course, knowing there are two solutions in each quadrant doesn’t help you figure out their values, but at least you can be sure they are there to be found.
Maybe a substitution in the quartic equation where you had x^4 and x^2. Set something like u=x^2 and solve with the quadratic formula for u. And then you can do it again for x?
Hi &E, love your math puzzles! Just wanted to point out that the answers could be simplified and the y values look different but are equal to the alternate x values. 36/√54 = √24 and 36/√24 = √54. Doing the math to prove that they are the same was kinda fun. But the possible solutions in simplest form are: X=2√6 Y=3√6 or X=3√6 Y=2√6 or X=-2√6 Y=-3√6 or X=-3√6 Y=-2√6 Not trying to be a know it all because I'm not, if your puzzle involves circles my solve rate is about 50% without hints, my trig is very rusty! I look forward to each new math puzzle, keep up the great content!
Sorry for bad English but it's a known condition for cases such as that one which let you simplify things but not everyone in the world learns about them. Like how some learn about the ABC formula but not PQ formula and vice versa.
(x+p)(x+q)=x²+px+qx+pq, pq is a constant with no x, therefore the two numbers we are trying to find, when multiplied, should equal to the constant. In this case, 1296. px+qx means that the coefficient of x is p+q, therefore p+q should be equal to the coefficient of x. E.g. factorisation of x²+2x-3, p+q = coefficient of x, which is 2 pq = constant, which is -3 You then do trial and error till you find the correct combination. In this case, the answer is (x+3)(x-1)
@@AkitoLitethanks, i really wasn't in the mood to think about it, but 2 people came in to help, so i forced myself to rewatch the video and think about it with your explanation i do feel like i understand it better, specially due to that example with x² +2x -3, the last part having (x+3)(x-1) made me feel like i maybe understand it now, thanks
@Andy Math I don’t know if it will be easier to use identity for this question since for this way it doesn’t nneed to find the solution of x and y and we can calculate to the final answer ?
I'm convinced that most of us watch Andy because we love math. But, us women (some men, too) watch Andy also because he's cute. Sorry that my comment is so long. (My husband and I quote Gene from Bob's Burgers all the time, "we're married, not buried." There is no shame in admiring beautiful people; that's why while rewatching Home Improvement I had to skip the episode where Jill gets super mad and jealous because Tim checked out another woman). 😆
So are you telling me that we’ve seen amazing videos like this for more than 7 years and we’ve never had a video to know more about Andy? Why Andy? Why?
As to solving this problem using a longer method, Good Job - easily understood and not really difficult. As to the Haircut, Good Work there, too! Although, I don't understand why someone with great hair needs to have a hat on indoors and in front of a camera? 😂😂😂
hey Andy great video, however I had an alternate solution to this, just square on both sides the equation x²+y²=78 and we'll have the answer without having to deal with square roots 😅
Hi! I dont really know where i am! I was wondering youtube and went into my youtube channel, and saw that you are a subscriber of mine :) How did that happen?
I would like to understand the p and w part more, I feel like when that started happening in math class it really took a lot of joy out of math for me because my brain does not comprehend
Sometimes factoring is too difficult or takes more time than to use the quadratic formula. The quadratic formula works here because if you subsitute x^2 for let's say u, then it becomes a quadratic equation. Then you can use the quadratic formula, and then you can substitute the x^2 back into u. Hope this is clear and helpful!
Whenever you have a formula that us written like: a^2 + 2ab + b^2 You can transform it into a format (a+b)(a+b) This is easy because b can always be found with root of b^2 The tricky part is knowing what the factors are when the formula is: a^2 + a(b + c) + bc Which is (a+b)(a+c) How is thus second method done you might ask? Well, lets make an example: x^2 + 10x + 16 1. First you look at which numbers factor up to x^2 - that's simple it's x - so we fill it in the formula: (x+b)(x+c) 2. You look at which numbers factor up to 16 - 4*4 = 16 - 16*1 = 16 - 2*8 = 16 - etcetera 3. Now you look at which of these would summ up to 10: - 4+4=8 - 16+1=17 - 8+2=10 4. so the formula is (x+8)(x+2) Now things can get more complex when using minus signs or fractions, but let's not get ahead of ourselves. You can test your skills online researching: trinomials or binomial products.
I want to add a solution for my side Add and subtract the eqn 1 by 2 x^2y^2 So the eqn become x^4+y^4 +2x^2y^2 -2x^2y^2 Then by identity a^2+b^2 +2ab= (a+b)^2 (X^2+y^2)^2 -2x^2y^2 Put the value (78)^2 - 2 (36)^2 6084- 2 (1296) 6084 -2592 3492
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
Fun fact: Andy is always handsome with any hairstyle
Yep. When he says "how exciting" I always think he is talking about himself.
I see why Andy left Toy Story
gay
@@burntsouffleoms 😭🙏
@@leodame3 omsimize
Let's get an hour long live stream of Andy solving fan submitted math problems!!!
YES PLS3AS3
Not just exciting, absolutely electrifying.
How exciting ❌
How absolutely electrifying ✅
Target audience 🧒
Actual audience 🧑💼
For real! I need none of this math. I'm an accountant. 😆
Lot of confusion in the comments about the p and q part.
Quick written explanation for anyone not sure what happened in that step, it can be reduced from a quartic to a simple quadratic with a substitution of a=x². Solving a² - 78a + 1296 = 0 using the quadratic formula gives the roots.
It might be worth noting that before we dive into the algebra, we can infer some properties of the solutions. We are trying to solve two simultaneous equations whose solutions correspond to the intersection of a circle of radius SQRT(78) and a hyperbola with vertices at (6,6) and (-6,-6); we know that any solutions will be mirrored in the first and third quadrants of the cartesian plane. We can also infer that since x and y are interchangeable in the equations, they must be so in the solutions (i.e., you should be able to swap the x & y values of a solution to get another solution - another way of saying the solutions are reflected in the line y = x). Finally, we can infer there must be 4 real solutions (two in each of quadrant 1 and quadrant 3), rather than zero or 2 (one in each quadrant), by noting that the circle’s radius is greater than SQRT(2*36), so it must intersect the hyperbola at two points in both quadrants (draw yourself a diagram of the two equations in the cartesian plane to see this). If the radius was less than SQRT(72), there would be no intersections (i.e., zero real solutions), and if it was exactly SQRT(72) (i.e., x^2 + y^2 = 72), then there would be exactly one solution in quadrant 1 and one in quadrant 3 (i.e., at the hyperbola’s vertices). Of course, knowing there are two solutions in each quadrant doesn’t help you figure out their values, but at least you can be sure they are there to be found.
That's a great illustration thanks!
HOW EXCITING 🔥🔥🔥
fire haircut🔥
Astounding reasoning again by Mr Andy 👍
How exciting🌟🤩
Short cut, long path. Whats the medium solution? Lol
Maybe a substitution in the quartic equation where you had x^4 and x^2. Set something like u=x^2 and solve with the quadratic formula for u. And then you can do it again for x?
x2 + y2 - 2xy= 78 - 2*36
(x - y)2 = 6
x - y = sqrt(6)
x = y + sqrt(6)
plug it into xy = 36, find x and y
Cool I found the same result but with a different way❤❤🎉. How exciting 🎉🎉🎉🎉
Neat solution and neat haircut.
Loveee the hair 😂
Love this guy!
Takes hat off.
I’ve got hat hair now, I don’t know why.
😂
Thank you, Andy.
math student: QED
andy math student: how exciting 🤩
How exciting !!!
Hi &E, love your math puzzles! Just wanted to point out that the answers could be simplified and the y values look different but are equal to the alternate x values.
36/√54 = √24 and
36/√24 = √54.
Doing the math to prove that they are the same was kinda fun.
But the possible solutions in simplest form are:
X=2√6 Y=3√6 or
X=3√6 Y=2√6 or
X=-2√6 Y=-3√6 or
X=-3√6 Y=-2√6
Not trying to be a know it all because I'm not, if your puzzle involves circles my solve rate is about 50% without hints, my trig is very rusty! I look forward to each new math puzzle, keep up the great content!
I do NOT get any of these videos but I still watch them 🥴
Same
i didn't really understand that part with p and q, i probably could if i think about it long enough tho
Sorry for bad English but it's a known condition for cases such as that one which let you simplify things but not everyone in the world learns about them. Like how some learn about the ABC formula but not PQ formula and vice versa.
(x+p)(x+q)=x²+px+qx+pq,
pq is a constant with no x, therefore the two numbers we are trying to find, when multiplied, should equal to the constant. In this case, 1296.
px+qx means that the coefficient of x is p+q, therefore p+q should be equal to the coefficient of x.
E.g. factorisation of x²+2x-3,
p+q = coefficient of x, which is 2
pq = constant, which is -3
You then do trial and error till you find the correct combination.
In this case, the answer is (x+3)(x-1)
@@AkitoLitethanks, i really wasn't in the mood to think about it, but 2 people came in to help, so i forced myself to rewatch the video and think about it
with your explanation i do feel like i understand it better, specially due to that example with x² +2x -3, the last part having (x+3)(x-1) made me feel like i maybe understand it now, thanks
HOW EXCITINGGGGF
Thats a pretty nice haircut
Is Andy the oldest 20yo or the youngest 40yo?
Maybe he is just 30......?
How exhilarating!
@Andy Math I don’t know if it will be easier to use identity for this question since for this way it doesn’t nneed to find the solution of x and y and we can calculate to the final answer ?
Hello Andy, I'm a fan of your content and I'd like to suggest a math problem:
fully simplify: 3^100 + 3^100 + 3^100/3^101-3^100-3^99
Well As far as I could go,
2×3^99 + 1/3
Andy math is the type of guy that gives my 10th grade math knowledge a purpose 😂
Greetings from Poland.
greetings from kenya
How exciting
“how exciting” indeed
😂 how exciting🎉
pov: you have not developed dynamic problem solving skills
which program you use to make that motions with the equations? I would love to use it in my classes
I'm convinced that most of us watch Andy because we love math. But, us women (some men, too) watch Andy also because he's cute.
Sorry that my comment is so long.
(My husband and I quote Gene from Bob's Burgers all the time, "we're married, not buried." There is no shame in admiring beautiful people; that's why while rewatching Home Improvement I had to skip the episode where Jill gets super mad and jealous because Tim checked out another woman). 😆
W haircut
Always exciting solutions!
Fans: no shortcut
Andy: I did use the calculator.
How exciting.
I'm from Brazil and i really liked your videos. What programs do you use to make these videos?
Mostly PowerPoint
The haircut was our present for watching the whole video
Maths are beautiful
Something unrelated to Math. Can you make a video saying “how exciting” on a loop? Haha
So are you telling me that we’ve seen amazing videos like this for more than 7 years and we’ve never had a video to know more about Andy? Why Andy? Why?
how exciting
As to solving this problem using a longer method, Good Job - easily understood and not really difficult. As to the Haircut, Good Work there, too! Although, I don't understand why someone with great hair needs to have a hat on indoors and in front of a camera? 😂😂😂
hey Andy great video, however I had an alternate solution to this, just square on both sides the equation x²+y²=78 and we'll have the answer without having to deal with square roots 😅
I think it can be solved using the Newton-Girard formulas
Worries about *multiplying* by x if it's zero (??), but has already blithely divided by x 🤔. The haircut makes up for everything though :-)
Hello Andymath
I always solved w the shortcut and thus never thought or tried the long* cut
How do I submit questions
Let a = x^2,b = y ^ 2.
Therefore,from what's given,
a + b = 78 __ (1)
And we knew that xy = 36
Therefore,x^2.y^2 = 36.36 = 1296.
Hence,
ab = 1296.
Now x^4 + y^4 = a^2 + b^2,
Also, a^2 + b^2 = (a+b)^2 - 2ab
Also,
a+b = 78,ab = 1296
Therefore,(a^2 + b ^ 2) = (78)^2 - 2 * 1296
= 6084 - 2592
= 3492.
Hi! I dont really know where i am! I was wondering youtube and went into my youtube channel, and saw that you are a subscriber of mine :) How did that happen?
Also, if x=+-sqrt(24) THEN y=+-sqrt(54) and vice versa ;).
czekolady w 3 min
Could you not have solved the quadratic equation (where x square equals z for example) instead of looking for p an q manually?
I noticed (x^2 + y^2)^2 = (x^4 + y^4) + 2(xy)^2. From there its simple substitution and algebra
To be honest, I wanted to see you show how you got p & q even though you used a calculator 😅
Andy math op😊😊
X^2 on top and bottom reduce to 1! Not cancel. They are not positive and negative charges.
Bro pls react and solve the que of iit jee advance maths
Guys, how can i send him a math problem?
Andy, give my comment a heart, pls
Andy how old are you
Can anyone sharethe link for the shortcut method for these type of problem
Hey Andy, I don't know if this is the place but do you have a degree in math?
2592?
I would like to understand the p and w part more, I feel like when that started happening in math class it really took a lot of joy out of math for me because my brain does not comprehend
I think there is a formula to solve it, but idk if it works
p+q = number 1
p=number1-q
(number1•q)(q) = number2
number1q q² = number 2
then solve it
Sometimes factoring is too difficult or takes more time than to use the quadratic formula. The quadratic formula works here because if you subsitute x^2 for let's say u, then it becomes a quadratic equation. Then you can use the quadratic formula, and then you can substitute the x^2 back into u. Hope this is clear and helpful!
Whenever you have a formula that us written like:
a^2 + 2ab + b^2
You can transform it into a format
(a+b)(a+b)
This is easy because b can always be found with root of b^2
The tricky part is knowing what the factors are when the formula is:
a^2 + a(b + c) + bc
Which is
(a+b)(a+c)
How is thus second method done you might ask? Well, lets make an example: x^2 + 10x + 16
1. First you look at which numbers factor up to x^2
- that's simple it's x
- so we fill it in the formula:
(x+b)(x+c)
2. You look at which numbers factor up to 16
- 4*4 = 16
- 16*1 = 16
- 2*8 = 16
- etcetera
3. Now you look at which of these would summ up to 10:
- 4+4=8
- 16+1=17
- 8+2=10
4. so the formula is (x+8)(x+2)
Now things can get more complex when using minus signs or fractions, but let's not get ahead of ourselves. You can test your skills online researching: trinomials or binomial products.
Leaving your y with a sqrt in the denominator hides the symmetry between x and y :(
You are right. That would have been better to show that.
Bro this is 8th grade problem damn.
no views, one comment and three likes.. youtube is drunk again.
true..
false...
Some people like and comment before they have watched enough of the video where RUclips will count it as a view.
You should tell to your barber to cut your hair without the shortcut.
2 and 3 for x and y and 97 for the answer? At time 0:00 , used the thumbnail to answer, no calculator
żb stratwgia lifestylu byla faktycz dobra
I just use x² =54/24 to get y²=24/54
And then just put this in equation to het 3492 why would you go to sqrt x²?
78^2 - 2(36^2)=3492
Much much easier solution...
x² + y² = 78
x⁴ + 2x²y² + y⁴ = 78²
2x²y² = 2x36²
x⁴ + y⁴ = 78² - 2x36²
-24 and 24 sum to zero!
how is that a shortcut?
Normal Comment
ruclips.net/video/k8pIAJeOPY4/видео.htmlsi=kzngn2K3YNzbUGtw
Is this even legal maths?
Nerd
Have you ever heard of the formula a+b whole square 😒.. stupid method
Too long. Short solution: 78^2 - 2* 36^2
It was too easy for me!
(x² + y²)² - 2(xy)² = x⁴ + y⁴
Put the values of x² + y² and xy
You will get your answer quickly
very long, shortcut method (a+b)^2 = a^2 + b^2 + 2ab here a,b are x^2 and y^2 respectively then (78)^2 = x^4 + y^4 + 2(36)^2
I want to add a solution for my side
Add and subtract the eqn 1 by 2 x^2y^2
So the eqn become
x^4+y^4 +2x^2y^2 -2x^2y^2
Then by identity
a^2+b^2 +2ab= (a+b)^2
(X^2+y^2)^2 -2x^2y^2
Put the value
(78)^2 - 2 (36)^2
6084- 2 (1296)
6084 -2592
3492