X+17=sqrt[10](x-4)

Question

How to prove that x+17=sqrt[10](x-4) have no solutions in the real numbers?

Answer 1

As written, the factor (x-4) is NOT part of the radicand, so squaring both sides of this equation gives

x² + 34x + 289 = 10(x² - 8x + 16) and further simplification gives

9x² - 114x - 129 = 0...unless I've erred.

 

The discriminant of the resulting quadratic equation is > 0 and therefore there must be two Real solutions.