Solve y^4 = y^2 - x^2 to obtain four separate implicit functions of the equation

Question

Use the quadratic formula to solve the equation y⁴-y²+x²=0 for y² in terms of x. Then take the square root of both sides to solve these two equations for y.

I haven't done the quadratic equation in years and have no idea how this equation fits into the quadratic formula.

Answer 1

y⁴ = y² – x²

Let u=y²

u² - u + x² = 0

By quadratic formula,

u = -[+1 ± ((1 -4x²))^1/2]/2

y² = -[+1 ± ((1 -4x²))^1/2]/2

y = (-[+1 ± ((1 -4x²))^1/2]/2)^1/2