Solved an AMAZING Radical Equation with Substitution!

Solved an AMAZING Radical Equation with Substitution!
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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:
• An Interesting Radical...
• An Amazing Radical Equ...
• An Interesting Radical...
• A Tricky Radical Equat...
#math #radicalequation #algebra #problemsolving #education #matholympiad #matholympiadpreparation #tutorial #quadraticequations
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[@gregevgeni1864]

The given equation written
[781 - (x -5)^5]^1/5 = 1 -(x -5) (*).
Put x -5 = y (**)
The (*) written (781 - t^5)^1/5 = 1 - t => 781 - t^5 = (1 - t)^5 =>
t^4 - 2 t^3 +2 t^2 - t- 156=0 =>
(t -4)(t +3)(t^2 -t + 13) = 0 (Horner).
So t = 4, t = -3 and two complex roots.
From (**) for t = 4 => x = 9 and
for t = - 3 => x = 2.
The x = 2 and x = 9 are accepted because both of them satisfy the given equation .

[@tejpalsingh366]

781+243= 1024
781-1024=- 243
Gives x= 2;9

[@RyanLewis-Johnson-wq6xs]

Surd[781-(2-5)^5,5]+2=6 Surd[781-(9-5)^5,5]+9=6 x=2 x=9 It’s in my head.

[@nitingl4730]

RHS =4+2 gives hint and clue

[@danmimis4576]

Nice, 2 real solutions and 2 complex, but shouldn't there be 5 solutions?

[@user-kt1dm9jz5t]

X1=2, X2=9.

[@kassuskassus6263]

x=2 and x=9

[@walterwen2975]

Solved an AMAZING Radical Equation: ⁵√[781 - (x - 5)⁵] + x = 6, x ϵ R; x=?
Let: y = x - 5, y ϵ R; No complex or imaginary value root
⁵√[781 - (x - 5)⁵] = 6 - x, ⁵√(781 - y⁵) = 6 - (y + 5) = 1 - y
781 - y⁵ = (1 - y)⁵ = 1 - 5y + 10y² - 10y³ + 5y⁴ - y⁵, 5(y⁴ - 2y³ + 2y² - y - 156) = 0
y⁴ - 2y³ + 2y² - y - 156 = (y⁴ - 2y³ + y²) + (y² - y) - 156 = (y² - y)² + (y² - y) - 156 = 0
(y² - y - 12)(y² - y + 13) = (y - 4)(y + 3)(y² - y + 13) = 0; y² - y + 13 ≠ 0, y = x - 5
y - 4 = 0, y = 4 = x - 5; x = 9 or y + 3 = 0, y = - 3 = x - 5; x = 2
Answer check:
⁵√[781 - (x - 5)⁵] + x = 6
x = 9: ⁵√[781 - (9 - 5)⁵] + 9 = ⁵√(- 243) + 9 = - 3 + 9 = 6; Confirmed
x = 2: ⁵√[781 - (2 - 5)⁵] + 2 = ⁵√1024 + 2 = 4 + 2 = 6; Confirmed
The calculation was achieved on a smartphone with a standard calculator app
Final answer:
x = 9 or x = 2

[@Fjfurufjdfjd]

Χ=9 ή Χ=2

[@RajeshKumar-wu7ox]

X=8,9 others roots are complex.

[@RajeshKumar-wu7ox]

X=2,9

[@RealQinnMalloryu4]

x^5 ➖ (x^5 ➖781 3125)+x=x^5 2➖(781 )^23120+x =x^5 ➖{55 561 ➖ 3120}+x ={x^5)^2 52.441+x={x^25 ➖ 52.441}+x=52. 416 +{x +x ➖ }={52.416+x^2}=52.416x^2 4^13.10^40^4^4x^2 4^13.102^20^4^4x^2 4^13.10 2^2^10^4^4x^2 4^13^1.2^5^1^12^5^2^2^2^2x^2 4^1^1.1^1^1^1^1^1^1^1x^2 2^2x^2 1^1x^2 1x^2 (x ➖ 2x+1).

[@Shobhamaths]

Let a^5=781-(x-5) ^5;
b=x-5
a^5+b^5=781;
a+b=1
a^5+(1-a) ^5=781
a^4-2a^3+2a^2-a-156=0
a=-3, 4
then x=2, 9 others are complex👍

[@user-kp2rd5qv8g]

Let x-5=y and [781-(x-5)^5]^1/5=z. Then, y+z=1 and y^5+z^5 = 781. If yz=a, y^3+z^3=1-3a as y+z=1. So, (y+z)^5=1 = y^5+z^5 + 5 yz(y^3+z^3) + 10y^2z^2(y+z) > a^2-a-156=0 > a=-yz=12,13. Ifyz=-12, y=4,-3 > x= 2,9. Both are valid solutions. If a=13, there is no real value for y. So, x=2,9.