An Interesting Polynomial System

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2x^2-3xy+y^2=3 and x^2+2xy-2y^2=6
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[suk-younsuh]

From your 2nd method, 3x^2 - 8xy + 4y^2 = 0, by substituting 4y^2 - 8xy = 4(y - x)^2 - 4x^2, you get 4(y - x)^2 - x^2 = 0, which is difference of two squares. Thus, you get: y = x and y = (1/2)x. The rest is straightforward.

[XJWill]

I think the easiest way is to just add multiples of the equations to get two new equations with the x*y and the y^2 terms eliminated:
2*{eq1} + 3*{eq2} : -4*y^2 + 7*x^2 - 24 = 0
2*{eq1} + 1*{eq2} : -4*x*y + 5*x^2 - 12 = 0
Now just solve the last for y and plug it into the previous equation
y = 5*x/4 - 3/x
3*x^2/4 - 36/x^2 + 6 = 0
That is quadratic in x^2 and we find
x^2 = 4 OR x^2 = -12
So the 4 roots for x are
x1 = 2
x2 = -2
x3 = sqrt(-12)
x4 = -sqrt(-12)
Then we go back to the equation for y - 5*x/4 - 3/x and get the corresponding y roots
y1 = 1
y2 = -1
y3 = sqrt(-27)
y4 = -sqrt(-27)

[bucc]

The other two solutions are (2i sqrt(3), 3i sqrt(3)) and (-2i sqrt(3), -3i sqrt(3)).

[SG]

2 comments: Number 1: 3rd and 4th solutions are simply complex, but they work as well.
Number 2: I did it the same way like you started in method 2, but once I had the new equation, I just used the quadratic formula to solve for y with considering x as a constant. I think it is not really necessary to do the substitution with t, it just complicates the method but does not really have any benefits. Luckily when we plug in the quadratic formula we get x^2-3/4x^2 under the square root, therefor it simplifies to x and we get y=x+/- 1/2*x, means either y=x/2, or y=3/2*x

[williamperez-hernandez]

For 2nd method we have the factorization: 3x^2 - 8xy + 4y^2 = (3x - 2y)( x- 2y) = 0, thus x = 2y and x = 2y/3.

[AmirgabYT]

I'm only who saw the six is drawn 😅?

[scottleung]

I used the second method, but without substitution.

[monkeblazer]

if u multiply first equation by 2 and equate, u can substitute and find value of x and y, ig this is an easier approach. ALSO SYBERMATH CAN U PLEASE UPLOAD OLYMPIAD PROBLEMS AS IOQM IS COMING UP AND IT WOULD BE HELPFUL FOR OUR PREPARATION