Germany | Can you solve this? | Math Olympiad

[@에스피-z2g]

(x-3)^4=16
x-3=2 or x-3=-2 or
x-3=2i or x-3= -2i
x=5 or x=1 or
x=3+2i or x=3-2i

[@vilastadoori]

X-3 to the power of 4 is equal to 2 to the power of 4 then X-3 = 2 and x = 5

[@chrys0gonus]

(x-3) is either 2 OR (-2), because (-2) to the power of 4 also equals 16.
So x is either 5 or 1.

[@IsraSirampa]

(x-3)⁴ = 16
(x-3)⁴ = 2⁴
x-3 = 2
x = 2 + 3
x = 5
--------------------
Subtitusi x = 5 ke persamaan ;
(x-3)⁴ = 16
(5-3)⁴ = 16
2⁴ = 16
16 = 16

[@chrys0gonus]

@@IsraSirampa substitusi x=1 ke persamaan:
(1-3)^4=16
(-2)^4=16
16=16

[@IsraSirampa]

@@chrys0gonus
(x-3)⁴ = 16
(x-3)⁴ = 4²
((x-3)²)² = 4²
(x-3)² = 4
(x-3)² - 4 = 0
x² - 6x + 9 - 4 = 0
x² - 6x + 5 = 0
(x - 1)(x - 5) = 0
(x - 1) = 0 ; (x - 5) = 0
x - 1 = 0 ; x - 5 = 0
x = 1 ; x = 5
HP {1 ; 5}

[@mengmardiati1345]

​@@IsraSirampaaku jg jawabnya spt ini

[@user-t-r-u]

5

[@stpat7614]

This way is easier:
(x - 3)^4 = 16
(x - 3)^(2 * 2) = 16
([x - 3]^2)^2 = 4^2
Let a = (x - 3)^2, and b = 4
a^2 = b^2
a^2 - b^2 = b^2 - b^2
a^2 - b^2 = 0
(a - b)(a + b) = 0
([x - 3]^2 - 4)([x - 3]^2 + 4) = 0
([x - 3]^2 - 2^2)([x - 3]^2 + 2^2) = 0
Let a = x - 3, and b = 2
(a^2 - b^2)(a^2 + b^2) = 0
(a - b)(a + b)(a - i * b)(a + i * b) = 0
([x - 3] - 2)([x - 3] + 2)([x - 3] - i * 2)([x - 3] + i * 2) = 0
(x - 3 - 2)(x - 3 + 2)(x - 3 - i * 2)(x - 3 + i * 2) = 0
(x - 5)(x - 1)(x - 3 - i * 2)(x - 3 + i * 2) = 0
x - 5 = 0, or x - 1 = 0, or x - 3 - i * 2 = 0, or x - 3 + i * 2 = 0
Suppose x - 5 = 0
x - 5 = 0
x - 5 + 5 = 0 + 5
x = 5
Suppose x - 1 = 0
x - 1 = 0
x - 1 + 1 = 0 + 1
x = 1
Suppose x - 3 - i * 2 = 0
x - 3 - i * 2 = 0
x - 3 - i * 2 + 3 + i * 2 = 0 + 3 + i * 2
x = 3 + i * 2
Suppose x - 3 + i * 2 = 0
x - 3 + i * 2 = 0
x - 3 + i * 2 + 3 - i * 2 = 0 + 3 - i * 2
x = 3 - i * 2
x1 = 5
x2 = 1
x3 = 3 + i * 2
x4 = 3 - i * 2

[@Mr_Basketball95]

Maybe de moivre would be a more general solution but nice approach

[@MasterTMathsClass]

✌️

[@NoKar0821]

Смысл какой в таком решении? Процесс ради процесса? Или за каждую строчку платят деньги? Сразу корень извлечь из двух частей уравнения.

[@chrys0gonus]

If (x-3)^4=16 then (x-3) has to be either +2 or -2. Thus, x=5 or x=1.
Solved in under 10 seconds… 🤷🏼‍♂️

[@b213videoz]

you lost half of the solution

[@audreydaleski1067]

4th root

[@ГригорийМоисеев-в4о]

А что мешало сразу корень извлечь?

[@b213videoz]

(x-3)⁴ = 2⁴ = (-2)⁴ = (-2i)⁴
x-3 = [2, -2, -2i]
x = [5, 1, -2i+3]
What did I miss ?

[@b213videoz]

Right! So I missed 3+2i

[@mehdizangiabadi-iw6tn]

x=5