A Very Nice Math Olympiad Problem | Solve for real value of x for which x^30+x^20=80

In this video, I'll be showing you step by step on how to solve this Olympiad Maths problem using a simple trick.

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[michaeledwards]

Somebody loves 64 + 16 = 80. (x^10)^3 + (x^10)^2 = 80. Initially x^10 = 4, giving x = 2^(1/5) with another 9 roots to find.

[mikeeisler]

2 problems

1. 4^(1/10) isn’t simplest form

4^(1/10)
= (2^2)^(1/10)
= 2^(2/10)
= 2^(1/5)

2. There are 2 real solutions:


x^10 = 4

x^10 - 4 = 0

(x^5)^2 - 2^2 = 0

Difference of squares

(x^5 - 2)(x^5 + 2) = 0

x = +- 2^(1/5)

[sy]

I would like to show the 2 points as follows:
1) There is another solution: x = - (4^(1/10)) [ There are 2 real solutions. ]
2) As for writing the final answer, 2^(1/5) is better than 4^(1/10) .
Therefore, my final answer is x = ± 2^(1/5) .
<<<< I solved it by mental arithmetic from beginning to end. >>>>