Y=3x⁵ solve for x

Question

Quadratic equation

Answer 1

Take the 5th root of each side:

⁵√y=x(⁵√3), x=⁵√(y/3).

This is the only real solution. There will be complex solutions (2 pairs of conjugate complexes).

Without going into specifics, the other 4 (complex) solutions for x can be presented as two conjugate pairs (i represents imaginary √-1):

(cos(72°)+isin(72°))(⁵√(y/3)), (cos(72°)-isin(72°))(⁵√(y/3));

(isin(36°)+cos(36°))(⁵√(y/3)), (isin(36°)-cos(36°))(⁵√(y/3)).

While real numbers can be thought of as points on a number line (one dimension), complex numbers can be thought of as points in a plane (two dimensions, the points being the vertices of a regular pentagon with radius=⁵√(y/3). The real solution is the point (⁵√(y/3),0).