How do you solve an equation with two variables 2x^2 - 3xy +2y^2 = 2

Question

i've tried to complete the square.

Answer 1

The diophantine equation

The term Diophantine Equation means that the solutions (x , y ) should be integer numbers.

Compare the above equation .

A = 2 , B = -3, C = 2, D = 0, E = 0, F = -2.

Discriminant

So the above equation is elliptical.

Elliptical case

Since the ellipse is a closed figure, the number of solutions will be finite.

For any value of x  there will be two values of y  except at the left and right extremes of the ellipse. In this case there will be only one value of y . To determine the location of the left and right extremes we should equal the square root to zero, so the previous expression returns only one value of y .

So the values of x should be between the roots of this equation.

Solutions

.

 

Comments

The value of y = (1/4) [3x - sqrt(16 - 7x²)] and the value of x = ±4/√7.

To determine the location of the left and right extremes we should equal the square root to zero, so the previous expression returns only one value of y .

Since sqrt(16 - 7x²) = sqrt[16 - 7(±4/√7)²] = sqrt[16 - 16] = 0.

The remaining y value is y= (1/4) 3x.

If x = 4/√7 then y = (1/4) [3 (4/√7)] ----> y = 3/√7.

If x = - 4/√7 then y = (1/4) [3 ( - 4/√7)] ----> y = - 3/√7.