A Nice Radical Equation Challenge | Algebra | Math Olympiad

A Nice Radical Equation Challenge | Algebra | Math Olympiad

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In this algebraic video, we explore an intriguing radical equation that’s sure to challenge our algebraic skills. This problem is an excellent exercise for anyone preparing for Math Olympiad or simply looking to deepen their understanding of advanced algebra. Follow along as we break down the steps to solve this complex equation, and try to solve it yourself before we reveal the solution. Perfect for students and math enthusiasts alike!

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📌 Topics Covered:

Radical equations
Algebraic manipulation
Problem-solving strategies
Quadratic equations
Quadratic formula
Substitution
Real solutions
Verification

Additional Resources:

#math #radicalequation #algebra #problemsolving #education #matholympiad #matholympiadpreparation #tutorial #quadraticequations

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[gregevgeni]

Trivial solution x = 0 .
Let x ≠ 0 .
Set a = (x^5 + 20•x^3)^(1/7) (*) and
b = (x^7 - 20•x^3)^(1/5) (**) .
From the initial equation we obtain
a = b (#).
Also (*), (**) => a^7 = x^5 + 20 •x^3 and b^5 = x^7 - 20•x^3 .
Summing the last two equalities we have
a^7 + b^5 = x^7 + x^5 =>
a^7 + a^5 = x^7 + x^5 ( ***).
Define the function f(x) = x^7 +x^5 .
Observe that f '(x) = x^6 + x^4 > 0 for all x ≠ 0 ..
So the f is increasing function and
thus the is 1 - 1 .
Therefore, from (***),
f(a) = f(x) => a = x (##) .
From (*) and (##)
x = ( x^5 + 20• x^3)^(1/7)
or x^7 = x^5 + 20 • x^3
or x^7 - x^5 - 20 • x^3 =0
or x^3 • ( x^4 - x^2 - 20) = 0 .
But x^4 - x^2 - 20 = 0 =>
(x^2 = -4 => x = ± 2i ),
(x^2 = 5 => x = ±√5 ).
And the trivial solution x = 0 too.
Solutions of the given equation are
x = 0, x = ±2i, x = ±√5.

[kassuskassus]

Real solutions are x= -sqrt5, x=0 and x=sqrt5.

[tanbw]

Are there no other roots? You have only considered y=x. You have not looked at the remaining part of the x, y equation.

[RyanLewis-Johnson-wqxs]

Surd[(x^5+20x^3), 7]=Surd[(x^7-20x^3), 5]
x = 0
x =± Sqrt[5]

[RealQinnMalloryu]

x^7+x^5+20x^3={ x7+x^7 ➖ }=x^14{x^5+x^5 ➖}=x^10 {x^14+x^10}=x^24+{20x^3+20x^3 ➖ }={x^24+40x^6}=40x^30 2^20x^3^10 2^2^10x^3^2^5 1^1^2^5x^3^2^5 1^1x3^2^1 x^3^2 (x ➖ 3x+2) . (x^7 )^2➖( x^5)^2 ➖ 20x^3 ={x^49 ➖ x^25} ➖ 20x^3=x^24 ➖ (20x^3)^2= {x^24 ➖ 400x^9}=400x^15.10^40x^15 10^2^20x^15 10^2^2^10x^15 10^1^1^10^x^3^5 2^5^2^5x^3^5^1 1^1^2^1x^3^1^1 2x^3 (x ➖ 3x+2).