China | A Nice Algebra Problem | Math Olympiad

Hello My Dear CZcams Family 😍😍😍
Hope you all are doing well 🥰🥰🥰
If you are enjoying my video about how to solve this math olympiad algebra problem then please Like and Subscribe my channel as it helps me a lot 🙂🙂🙂

[@tejpalsingh366]

Sheer stpidity
Put x+9= t n then n3xt

[@unnevenn]

schlawg?

[@Irtsak]

① The substitution x+10 =t is brilliant .
In 8:24 Math Booster found : t⁴+6t²-352=0.
Let t²=w and then you have to solve a very easy equation.
w²+6w-352=0
Personally I use the discriminant formula.
D=36+4•352=1444=38²>0
So w=(-6±38)/2 => w=16 or w=-22 => t²=16 or t²=-22 ………..
② Another approach (more difficult):
(x+9)⁴+(x+11)⁴=706
(x+9)⁴+(x+9+2)=706 (let x+9=t)
=> t⁴+(t+2)⁴=706 ……..
t⁴+4t³+12t²+16t-345=0 (1)
The possible integral roots of the equation are the divisors of -345.
Hence : ±1,±3,±5,±23
*Using Horner’s method*
1 4 12 16 -345 ρ=3
# 3 21 99 345
1 7 33 115 0
So (1) => (t-3)(t³+7t²+33t+115)=0 (2)
1 7 33 115 ρ=-5
# -5 -10 -115
1 2 23 0
(2)=> (t-3)(t+5)(t²+2t+23)=0 ….. etc