Solve for both x and y

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You cannot find the unique values of two variables with only one equation. You can only establish a relationship between them.

Let z=y², then 3x⁴-7xz-2z²=4830091, which can be written:

z²+(7x/2)z=(3x⁴-4830091)/2,

z²+(7x/2)z+49x²/16=(3x⁴-4830091)/2+49x²/16,

(z+7x/4)²=(8(3x⁴-4830091)+49x²)/16,

z+7x/4=±√(24x⁴-38640728+49x²)/4.

z=y²=(-7x±√(24x⁴+49x²-38640728)/4,

y=±√(√(24x⁴+49x²-38640728)-7x)/2.

This shows the relationship of y in terms of x and a graph of the relationship resembles a hyperbola which crosses the x axis at about ±35.62. Between these x limits, the curve doesn’t exist, so there are no values of y. Otherwise, for x>35.62 or x<-35.62, there are two values of y for each single value of x. All points on the curve are solutions to the equation.

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