Can You Crack This Radical Equation? | An Algebra Challenge

Can You Crack This Radical Equation? | An Algebra Challenge

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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! Let's check if you are among top 10%.

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📌 Topics Covered:

Radical equations
Algebraic manipulation
Problem-solving strategies
Cubic equation
Factorization
Substitution
Quadratic equations
Quadratic formula
Exponent laws
Real solutions
Verification

Additional Resources:

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

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[RashmiRay-cy]

Let y = (9-x^3/2)^2/3. Then, x+y =5 and x^3/2+y^3/2 = 9. Let x^1/2+y^1/2 = an and x^1/2 y^1/2 = b. Then x^3/2 + y^3/2 = a (5-b) = 9 and a^2 = 5 + 2b > b = 1/2(a^2-5). So, eliminating b, we get a^3-15a+18=0. Note that a, b are positive. Thus, the only solution is a=3 > b = 1/2(9-5) = 2. Thus, x^1/2 = 1, 2 > x=1, 4. These are both valid solutions.

[souzasilva]

O difícil nessas questões é a escolha adequada para a mudança de variáveis. (The difficult thing about these issues is the appropriate choice for changing variables.)

[АндрейПергаев-зн]

На 3:00
Проще сделать замену
Sqrt x+sqrt y=a
sqrt (xy) =c
Тогда
х sqrt x+ysqrt y=(sqrt x+sqrt y) *(x+y-sqrt (xy) =a*(5-c)=9
И
х+у=(sqrt x+sqrt y) ^2-3*sqrt (xy) =a^2-2c=5
Дальнейшее просто

[gregevgeni]

<=> ( 9-x√x)^2/3 = 5-x <=>
(9-x√x)^2 = (5-x)^3 <=>
81 -18x√x + x^3 = 125-75x+15x^2-x^3 <=> 2x^3 -15x^2 +75x -18x•√x -44=0 (*)
Set √x =t . Then the (*)
2t^6 -15t^4 +75 t^2 - 18t^3 -44=0 =>
t = 1 or t =2. (Horner)
So x =1 or x=4.

[ronbannon]

For those interested in the real solutions, three exist: 1, 4, and (10+sqrt(66*sqrt(33)-318))/4.