Finnish high school math problem.

[Lion_King]

Homework Statement

x¹⁰⁰⁰ = x(2x)⁵⁰⁰-2⁹⁹⁸x²


Okay. This is difficult. It is easy to notice that the solutions are x=0 and x=2 but how do you prove that? I have tried to resolve it in different ways with no luck at all. Maybe I haven´t noticed something. Any ideas? This is Finnish final exam puzzle from 1989...

 

[mutton]

With high exponents and common bases, the trick is usually to factor.

First, expand $$(2x)^{500}$$ and then move all terms to one side. Factor out $$x^2$$ and note that x = 0 is a solution. What remains is a quadratic, which can be factored to get x = 2 as the only other solution.

 

[Lion_King]

How do you know that it is clever to expand (2x)⁵⁰⁰? For example, I tried to expand x².

Is this the correct way?
x¹⁰⁰⁰=x(2x)⁵⁰⁰-2⁹⁹⁸x²

(2x)⁵⁰⁰x¹⁰⁰⁰=(2x)⁵⁰⁰x(2x)⁵⁰⁰-(2x)⁵⁰⁰2⁹⁹⁸x²

2⁵⁰⁰x¹⁵⁰⁰-2¹⁰⁰⁰x¹⁰⁰¹+2¹⁴⁹⁸x⁵⁰²=0

x²(2⁵⁰⁰x¹⁴⁹⁸-2¹⁰⁰⁰x⁹⁹⁹+2¹⁴⁹⁸x⁵⁰⁰)=0
Okay, clearly x = 0 is a solution. But what´s the difference? There is still that exponent monster. How can I get that x = 2? I am not following...