Higher-degree rational equations

A product is zero when at least one factor is zero. Thus x − a = 0 or x − b = 0, giving x = a or x = b. Do not count a repeated root twice.

For v > 0, x⁴ = v has two roots ±⁴√v. The odd-power equation x³ = v has one real root ∛v. For v = 0, both have the single root 0.

For v > 0, x⁴ = v has two roots ±⁴√v. The odd-power equation x³ = v has one real root ∛v. For v = 0, both have the single root 0.

Let t = x + 1/x and square it. Include the cross term. Since x is positive, the desired sum is positive.

Graphical solutions of f(x) = a are the x coordinates where the graph meets y = a. Algebra often gives exact values while a graph generally gives approximations. Check every intersection and the specified interval.

The intersection quadratic discriminant D determines the number of real intersections: D > 0 gives two, D = 0 one and D < 0 none.

Denominator x − a must be nonzero, so x ≠ a. Keep this restriction throughout the transformations and check the final roots.

The area gives x(x − a) = b, or x² − ax − b = 0. Solve the quadratic and keep only x values for which both x and x − a are positive.

Square the desired sum and relate it to both given equations. Positivity determines the root sign.

Set t = x², requiring t ≥ 0. Solve t² − st + p = 0. Discard negative t; each positive t gives x = ±√t, and t = 0 gives only x = 0.