Germany | A Nice Math Olympiad Problem | X=? & Y=? 

Let √x=a, √y=b , a,b>0 a+b=10 ab=10 then....

Very good.

Once you got x + y = 80，xy = 100， then x and y are two roots of the following eq. z^2 - 80z + 100 = 0 z = (80 ± √(6400 - 400)) / 2 z = (80 ± √6000) / 2 z = (80 ± 20√15) / 2 z = 40 ± 10√15 So，x = 40 + 10√15，y = 40 - 10√15， or x = 40 - 10√15，y = 40 + 10√15.

4:58 in my country it is considered an error if the fraction sign is not written at the level of the equal sign

i ❤ Mathematics

You seem amazing if you solving it by complex analysis method, can you solving it sir?

Simple solution: Substitute √x=5+z and √y=5-z into the second equation and rearrange to z^2=15, which has roots z=±√15 Thus x=(5+z)^2=(5±√15)^2=40±10√15 and y=(5-z)^2=(5∓√15)^2=40∓10√15 Check: √x√y=(5±√15)(5∓√15)=25-15=10

No se puede hacer así. Se utiliza una de las ecuaciones dos veces. En el punto 2, utiliza la primera y la segunda operación

In my world letters can never equal numbers.

Are x and y only real numbers?

👏👏👌👌en ese caso cuándo y=+ , x=-, e inversa. EXCELENTE

Sorry, but √(40 + 10√15) and √(40 - 10√15) are not the right answers. The answers are: | √x + √y = 10 | | √(xy) = 12 ---------- R É P O N S E ---------- | √x + √y = 10 | | √(xy) = 12 | √x + √y = 10 | | √x·√y = 12 get: • a = √x • b = √y | a + b = 10 = S (as Sum) | | ab = 12 = (P as Product) recall: roots of x² - Sx + P = 0 are a and b x² - 10x + 12 = 0 Δ = (-10)² - 4·1·12 = 100 - 48 = 52 √Δ = ±√52= ±2√13 root #1: x = a = (-(-10) + 2√13)/(2·1) = (10 + 2√13)/2 = 5 + √13 • √x = a = 5 + √13 • √x = 5 + √13 • x = (5 + √13)² • x = 25 + 10√13 + 13 • x = 38 + 10√13 • x = 2(19 + 5√13) root #2: x = b = (-(-10) - 2√13)/(2·1) = (10 - 2√13)/2 = 5 - √13 • √y = a = 5 - √13 • √y = 5 - √13 • y = (5 - √13)² • y = 25 - 10√13 + 13 • y = 38 - 10√13 • y = 2(19 - 5√13) ----- final results ----- ■ x = 2(19 + 5√13) ■ y = 2(19 - 5√13) ----- check ----- x = 2(19 + 5√13) ≈ 74.055512 y = 2(19 - 5√13) ≈ 1.944487 | √x + √y = 10 | | √(xy) = 12 | √74.055512 + √1.944487 ≈ 9.999999 ≈ 10 | | √(74.055512·1.944487) = 12 🙂

Not very smart to use x as a variable and a multiplier.

banale