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China | Can you solve this? | Math Olympiad

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Explore the interesting world of exponential equations with varying bases and unlock the power of exponentiation! In this educational video, we delve into the concept of exponential equations where the base is not a constant value. Learn the fundamentals, discover the properties, and master solving equations with different bases, all explained clearly and concisely. Join us on this exponential journey and level up your mathematical prowess!

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Alamaths

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Alamaths

√2 + √x = 2 → where: x ≥ 0
√x = 2 - √2
(√x)² = (2 - √2)²
x = 4 - 4√2 + 2
x = 6 - 4√2

key_board_x

Thank you for explaining. The solution is (only) x=6-4√2 . [ x=6+4√2 must be rejected. ] Please check.

sy

I agree The solution is only x=6-4√2 . I had a great deal work to verify this solution in the original equation in an exactly way.

HansTelerud

x^0.5 = 2 - 2^0.5
Square both sides
x = 4 - 4.2^0.5 + 2
x = 6 - 4.2^0.5

habeebalbarghothy

Through this math you are trying to express yourself that what a scholar

prabirsarkar

The first step I would make is moving sqrt(2) to the right. This puts variables on one side, constants on the other. Then square both sides resulting in:

sqrt(x)^2 = (2 - sqrt(2))^2.

x = 2^2 - 2(2*sqrt(2)) + sqrt(2)^2.

All that's left is collecting terms and simplifying.❤

jamesharmon

I solved a little different:
√2 + √x = 2
√x = 2 - √2
(√x)^2 = (2 - √2)^2
x = 4 - 4√2 + 2
x = 6 - 4√2
There are only this solution for "x" in this problem.

toninhorosa

sqrt(x) = 2 - sqrt(2) >> x = 0.343 (elementary school)

jmlfa

Impressed the way you present math to beginners. Great job 👏
Speaking from India.

devapriyaguharoy

√x+√2=2
X=(2-√2) ^ 2= 4+2-2×2√2
=6-4√2=2(3-2√2)

lakshmikutty

√x=2--√2>>>square both sides.etc.

KhinMaungSan-qcuv

Too many unnecessary steps. Sol. 6+4√2 is incorrect.
√2+√x=2
√x=2-√2
x=(2-√2)^2
x=4-2*2√2+2
x=6-4√2

akr

√x=2-√2 ; i, e x=4+2-4√2 ; x=6-4√2 2nd method. √2+√x=2 ; i, e 2+x+2√2x=4 ; i, e x+2√2x=2 ; 2√2x=2-x ; 8x=x^2+4-4x ; x^2-12x+4=0 ; x= (12+√144-16)/2 or x= (12-√144-16)/2 ; i, e x= (12+8√2)/2 ; x= 6+4√2 or x= 6-4√2

subratabiswas

if the final answer is an irrational number, there is no point in doing the transformations. You can simply calculate in the first calculus by taking 2 from the radical

gagik

Redondeado
x = 0, 343...es irracional

yanssala

Siempre cuando obtengas un número al cuadrado, como lo haces, pierdes información crucial sobre el número que elevas al cuadrado. Esa información es la positividad o negatividad del número.
Para ser precisos, en el momento que elevas al cuadrado a 2 - x, estás permitiendo que 2 - x sea negativo y esto no puede ser.
Es por eso que obtienes dos soluciones a la ecuacion cuadrática, que no tienen por qué ser soluciones de la ecuación original. De hecho desde la ecuación original tenemos que solo habrá una solución obtenida como alguien te comentó antes.
x = sqrt(2 - sqrt(2)).

martinzavalaleon

Sorry to bother, but this is overly complicated (and false). By squaring an equation you can create extraneous solutions (you did! x=6+4√2 is false).
Much easier would be to just substract √2 from both sides and only then to square the equation. This gives directly x=6-4√2 which is the only solution.
The reason behind this is the "dichotomy" of squares and roots:
√4 has only one(!) solution: "2".
But x²=4 has two solutions: +√4 = +1 * 2 = "2" and -√4 = -1 * 2 = "-2"

zaj

1.bərabərliyin hər iki tərəfini 2-yə bölüb qüvvətə yüksəltdikdən sonra 4-ə vurmaq nəyə lazım idi?Elə birbaşa qüvvətə yüksəltsə idiniz eyni nəticə alınacaqdı.
2. X^2-12X+4=0 şevrilmiş kvadrat tənlik olduğundan onun köklərini
X=-P/2+-*(P/2)^2 -q kimi həll etsə idiniz daha əlverişli olmazdımı?
Təşəkkürlər.

Nazimİsmayılov-eu

This is easy math, not Olympiad level

edwinpittomvils

China | Can you solve this? | Math Olympiad