Olympiad Math | Solve Advance Algebric Expression in Simple Way

Olympiad Math | Solve Advance Algebric Expression in Simple Way

Hi Friends,

Today I will teach you tips and tricks to solve the given olympiad math question in a simple and easy method.

Learn how to solve the Olympiad Question quickly with my tips and tricks.

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[pietergeerkens]

Nice problem and solution.

Also, after solving the quadratic, note that
(x-8) = 3 ± √10
is a unit in
ℤ[√10];
thus it's reciprocal is it's conjugate at any integer power.

Done.

[jflamingo]

solved with quadratic formula and substitution for same result, just with either positive or negative answer. nice question, and very interesting solution 👍👍

[jorgepinonesjauch]

Logré resolverlo de una manera más sencilla, haciendo un trabajo previo completando un cuadrado de binomio (x-8)^2=6x-47 dónde me quedo que (x-8)^2 es igual 11+ raíz de 10, este valor lo reemplazo en 6x-47 lo que me da 19-6raiz de 10 de ahí racionalizo 1 partido por esa expresión, hago reducciones y llegó a 12raiz de 10

[에스피-ht]

P=required question
P=A-1/A
x^2-22x+111=0
x=11+sqrt 10 #1
x-8=3+sqrt 10
A=(x-8)^2
=19+6sqrt10
1/A=19-6sqrt10
P=A-1/A=12sqrt10
x=11-sqrt10 #2
The result is same as #1

[КатяРыбакова-шд]

X^2-22X+111=0 => (X-11)^2=10, => X-8=3+-V10. => (X-8)^2=19+-6*V10 => 1: (X-8)^2=19-+6*V10. => (X-8)^2 + 1: (X-8)^2= 19+19= 38. Это моё решение.

[rajeevn.v.]

Given equation is (x-11)+-@^2. x equals 11.

[prabhudasmandal]

If (x--8) =a, 1/(x--8) will be 1/a.so, the expression is ( a^2--1/a^2) I. e (a+1/a) (a--1/a). We already have( a--1/a) =6. Now(, a+1/a) =√{6^2+4=√40=+-2√10{applying conversion formula (a+b) to((a--b) and vice versa. So, the result is +-6×2√10=+-12√10.in case of integers only 12√10.

[advaithkumar]

just rewrite the quadratic as (x-8)^2 - 6(x-8) - 1 = 0 => x-8 - 1/(x-8) = 6 and proceed as above

[echandler]

When you took square root of (a+b)^2 = 40 to give (a+b)=sqrt(40), how did you know the sign was +? From the first equation you showed that (x-8)(x-14)=1. Thus the desired quantity is (x-8)^2 - (x-14)^2 = (x-11+3)^2 - (x-11-3)^2 = 4(x-11)*3 = 12(x-11). Completing the square in the first equation gives (x-11)^2 = 10. I know the correct answer is 12sqrt(10). How to justify this choice of sign?