Solving a Nice Polynomial System in Two Ways

If you need to post a picture of your solution or idea:
#ChallengingMathProblems #PolynomialEquations #PolynomialSystems
via @YouTube @Apple @Desmos @NotabilityApp
@googledocs @canva

PLAYLISTS 🎵 :

[damiennortier]

Here's a fonctionnal method :
Add 1 to both sides and set each equation equal to the other to got (x+1)^2 + y = (y+1)^2 + x
(x+1)^2 - x = (y+1)^2 - y
Set f(x) = (x+1)^2 - x = x^2+x+1
So you got f(x) = f(y)
Since f'(x) = 2x + 1, and - 1/2 is the critical point, we can say that x = y or x and y are can of like opposite by -1/2.
So x = - 1/2 - a and y = b - 1/2
Replace in the equation (x+1)^2 + y = (y+1)^2 + x
[(-1/2 - a) + 1] ^2 + (b - 1/2) = [(b - 1/2) + 1] ^2 - (1/2 - a)
(1/2 - a)^2 + b = (b + 1/2)^2 - a
1/4 - a + a^2 + b = b^2 + b + 1/4 - a
a^2 = b^2
So a = b or a = - b
What it suppose to mean?
Either x = - b - 1/2 and y = b - 1/2 or x = b - 1/2 = y
Substitute each solution in one of the first equations :
First case : x = y
x^2 + 3x = 4
x^2 + 3x - 4
x = [-3 +- sqrt (25)]/2 = (-3 +- 5)/2
So x = - 4 = y or x = 1 = y
Second case : x = - b - 1/2 and y = b - 1/2
(-b - 1/2)^2 + 2(-b - 1/2) + (b - 1/2) = (b + 1/2)^2 - 2b - 1 + b - 1/2 = b^2 + b + 1/4 - b - 3/2 = b^2 - 5/4 = 4
b^2 = 21/4
So b = (sqrt 21) / 2 or b = - (sqrt 21) / 2
So x = - (sqrt21 + 1)/2 and y = (sqrt 21 - 1)/2 or x = (sqrt21 - 1)/2 and y = - (sqrt21 + 1)/2
And all without seeing the video 😁

[richardfarrer]

One small shortcut in the second approach is to write it as x^2 + x + (x+y) = 4. Then replace (x+y) with -1.

[juanmolinas]

Good video syber, Nice fact to learn it... "(x-1)" is a factor if...

[kinshuksinghania]

The way I see it, we can also obtain the solution by the intersection of the conic curve x² + 3x + 3y + y² = 8 (which is a circle) and the line y = x. There must be 2 more solutions which can be obtained by intersection of x² + 3x + 3y + y² = 8 and x + y + 1 = 0

[michaelpurtell]

Pretty sure never would have considered first method as second approach was obvious however glad to know of the first as an additional tool

[wuhyltw]

Since factorization can be done without difficulty, I think the problem maker wanted to solve it easily without increasing the degree in the 2nd method. Those who are familiar with it may be able to find (1, 1), (-4, -4) in mental arithmetic. If you can do mental arithmetic to the last two pairs, it will be even greater.

[dvd]

I LOVED IT!!😃👍🏼
I could solved it by myself, by the first method, but second one is easier and more fun. Thanks!

[crustyoldfart]

Nice problem.
f(x)= 4 - x^2 - 2*x ; g(x)= 4 - f(x)^2 - 2*f(x) ; 
f(-4) = g(-4), f(-3) = g(-3), f(1) = g(1), f(2) = g(2) ;
A geometric illustration of intersecting values of a parabola and a quartic curve.

[mohamedabdelkaderahmed]

Many thanks for your video, I solved it before watching the complete solution to the end & I got the same solution 🙂

[sberacatalin]

Bun exercitiu dar direct metoda 2! 👍🇦🇩😉

[stompzy]

add them together and you get a
circle equation, from there you can start to calculate integer values

[j.r.]

I don't know whether I should feel clever or thick for having solved this problem by using the method you should not be using to solve a problem. :)

[SuperYoonHo]

i love these questions! Where is your source of these questions?

[rakenzarnsworld]

Let's pretend x=1
Then, 3+y=4
1+y^2+2y=4
That means, y=1
Solution: (1, 1)

[musicsubicandcebu]

Since it's clear that x=y, then the y in first equation can be switched for x giving x² + 3x -4 = 0 . . . and solutions x =1, and -4. How come there's a third solution? We don't need one, why go looking for trouble!
But seriously, where is my error?

[GourangaPL]

could it be solved by assuming y = kx?