Japanese | Can you solve this ? | Math Olympiad X=? & Y=?

[@kurtmager1626]

No need for the quadratic formula. Just use identities. (x+y)²=(x-y)²+4xy So x+y=+/- √((x-y)²+4xy) x,y=0.5((x+y)+/-(x-y))

[@ffggddss]

x - y = 4; xy = 4 Find x & y. Here's a trick we learned in high school algebra, back in the day, that will solve this problem. In a quadratic equation of the form x² - Sx + P = 0 the sum of its roots is S, and their product is P. [To see this, expand (x-r₁)(x-r₂) = 0, which obviously has roots r₁ and r₂.] In the current problem, we have the difference and product, not the sum and product; but no sweat - the difference is the sum of x and -y. So just negate the 2nd given equation, and write: x + (-y) = 4; x(-y) = -4 and now we know that the quadratic z² - 4z - 4 = 0 has roots z = x, -y. Easy to complete the square by adding 8 to both sides: z² - 4z + 4 = 8 z - 2 = ±2√2 z = 2 ± 2√2 Now we know that the product xy is positive, so x,y are either both positive (Case I) or both negative (Case II). Case I: x = 2 + 2√2, y = -(2 - 2√2) = -2 + 2√2 Case II: x = 2 - 2√2, y = -(2 + 2√2) = -2 - 2√2 Check these in the original, and they work. Fred

[@ffggddss]

When the video gets to 1:30, with y² + 4y - 4 = 0, the quadratic formula isn't what you want, especially in a time-limited test situation. Much faster is to complete the square by adding 8 to both sides.

[@zvonimirkujundzic6867]

Thanks for another clear example and simple explanation, becouse the most ordinary people dont know how to solve it!🎉

[@Ms_Math1]

Good

[@wagnerperlino5048]

Well done !

[@mircoceccarelli6689]

Observatio : 4y + y^2 = 4 y^2 + 4y + 4 = 8 ( y + 2 )^2 = 2 × 2^2 ( y + 2 + a )( y + 2 - a ) = 0 , a^2 = 2 😊👍👋

[@oahuhawaii2141]

It's easy to brute force this problem: x - y = 4 ... y = x - 4 x*y = 4 x*(x - 4) = 4 x² - 4*x + 4 = 4 + 4 x = 2 ± 2*√2 y = x - 4 = -2 ± 2*√2 (x, y) = (2*(√2+1) , 2*(√2-1)), (-2*(√2-1), -2*(√2+1))

[@bobbyheffley4955]

I got the y values by completing the square.

[@alamshaikhahmad2415]

(X_y=4----- x+y=4----'xy=40)

[@사현-w2o]

Thank you so much. 😊

[@oahuhawaii2141]

09:00 Not a perfect square.

[@АннаГригорьева-у8м]

Этот пример не похож на олимпиадное задание!!

[@MRBROGFOWN]

Results x = 2 + 2√2 y = 2√2 - 2

[@oahuhawaii2141]

There are 2 solutions.

[@EustahijeMihajlović]

Whats this kindergarden olympiad?

[@zvonimirkujundzic6867]

Becouse of kindergarten questions!😂

[@ffggddss]

Yeah, unless it's one of dozens of problems to solve in a limited time, so to do well, you have to be quick with the solution to each question. Fred

[@baovu8681]

X=2,Y=2

[@Vajtosan]

2-2 !=4