Question

2x^2 - 3y^2 = 5xy - 5

Answer 1

2x^2-5xy-3y^2=

=2x^2-6xy+xy-3y^2=

= (2x^2-6xy)+(xy-3y^2)=

=2x(x-3y)+y(x-3y)=

=(x-3y)(2x+y)

Answer 2

To analyze the given equation 2x^2 - 3y^2 = 5xy - 5, let's rearrange it to bring all terms to one side of the equation:

2x^2 - 3y^2 - 5xy + 5 = 0

This is a quadratic equation in two variables, x and y. Quadratic equations are usually represented as:

ax^2 + bxy + cy^2 + dx + ey + f = 0

In our equation, a = 2, b = -5, c = -3, d = 0, e = 0, and f = 5.

This quadratic equation can represent different types of conic sections, such as ellipses, parabolas, or hyperbolas, depending on the discriminant (b^2 - 4ac).

For our equation, the discriminant is:

b^2 - 4ac = (-5)^2 - 4 * 2 * (-3) = 25 + 24 = 49

Since the discriminant (49) is positive, this equation represents a conic section, and specifically, it represents a hyperbola.

To find the exact properties of the hyperbola, we need to convert the equation to its standard form, which is generally represented as:

((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1

where (h, k) is the center of the hyperbola, and 'a' and 'b' are positive constants related to the shape and size of the hyperbola.

To convert the equation 2x^2 - 3y^2 - 5xy + 5 = 0 into the standard form, we need to perform a process called "completing the square." However, this process is relatively complex for a hyperbolic equation with a mixed xy term like this one.

If you have specific values of 'x' and 'y,' we can find out if they satisfy the equation. However, without further information or specific values, it's challenging to provide more detailed information about the hyperbola represented by the equation.