Thailand Junior Math Olympiad Problem - Algebra

Thailand Junior Math Olympiad Problem - Algebra

[vks_quily]

So proud if I found the answer with the same approach !

[devondevon]

x+ 4/x = 5 answer
A different appraoch.
let n=sqrt x then
n^2-4/n^2 = n + 2/n
n^4 - 4 = n^3 + 2n (multiply both sides by n^2)
(n^2 -2)(n^2+2) = n(n^2 + 2) factoring
(n^2 -2) x 1 = n x 1
n^2-2 - n=0
(n-2)(n+1) =0
n=2 and n = -1
hence sqrt x = 2 and sqrt x = -1
hence x =4 and x = 1
plug in both values into x+ 4/x = 5 gives
using 4 gives:
4 + 4/4 = 5
using 1 gives:
1 + 4/1 = 5
answer =5

[titonumet]

Here's another approach :D
Square both sides and we have
(x - 4/x)^2 = (√x + 2/√x)^2
resolve the squares
x^2 - 2.x.(4/x) + 16/x^2 = x + (2.√x.2)/√x + 4/x
some algebra
x^2 + 16/x^2 - 8 = x + 4/x + 4
we substract 4 on both sides
x^2 + 16/x^2 - 12 = x + 4/x
we add and susbtract 8 on left side so we can complete square with x^2 + 16/x^2
(x^2 + 16/x^2 + 8) - 12 - 8 = x + 4/x
we have a perfect square trinomium
(x + 4/x)^2 - 20 = x + 4/x
now we substract x + 4/x on both sides
(x + 4/x)^2 - (x+4/x) - 20 = 0
now let u = x + 4/x and we have
u^2 - u - 20 = 0
with u = 5 and u = -2 the solutions, but we only take u = 5 since x + 4/x is positive
then x + 4/x = u = 5

[tontonbeber]

Brute forcing also works easily
By squaring both sides and multiplying by x2, we have the quartic x4 - x3 - 12x2 - 4x + 16 = 0
Roots are easy to find by trying integers and then dividing the polynom ...
Roots are -2, -2, 1, 4
The double root -2 should eventually be rejected in R because of sqrt
1 doesn't work as both members are of different sign before squaring
x=4 is then only solution while keeping everything in R and x + 4/x = 5

[prabhudasmandal]

The other way is, simplifying the given value we get(x^2--4) /x=(x+2) /√x or(x+2) (x--2) /√x*√x={x+2) /√x or dividing it is(x__2) =√x. Squaring and adjusting it isx^2--5x+4=0 or (x--1) (x--4) =0 or x=4, 1.so, x+4/x is 5. Yours is however, tricky.

[KrishnaVamshi.]

Professor please tell me how to become god at maths like you.

[chengxiliu]

This is below the average difficulty of normal Chinese high school level