Math Interview Tricks: From Stanford, Harvard, Cambridge to Oxford University✍️🖋️📘💙

[@raghvendrasingh1289]

xy & x - y are roots of equation
t^2 - 7 t+6 = 0
t = 1 , 6
Case 1
xy = 1 , x - y = 6
x^2 - 6 x - 1= 0
x = 3+√10 , 3 - √10
y = - 3+√10 , - 3 - √10
Case 2
xy = 6 , x - y = 1
x^2 - x - 6 = 0
x = 3 , - 2
y = 2 , - 3
hence there are four solutions
(3 , 2) , ( - 2 , - 3)
(3+√10 , - 3 +√10) & ( 3 - √10 , - 3 - √10)

[@key_board_x]

x²y - xy² = 6
xy.(x - y) = 6 → given: x + xy - y = 7 → x - y = 7 - xy
xy.(7 - xy) = 6 → let: xy = a
a.(7 - a) = 6
7a - a² = 6
a² - 7a + 6 = 0
Δ = (- 7)² - (4 * 6) = 49 - 24 = 25
a = (7 ± 5)/2 → recall: xy = a
xy = (7 ± 5)/2
First case: xy = (7 + 5)/2 = 6
x + xy - y = 7
x + 6 - y = 7
x - y = 1
y = x - 1 → restart:
xy = 6
x.(x - 1) = 6
x² - x = 6
x² - x - 6 = 0
Δ = (- 1)² - (4 * - 6) = 25
x = (1 ± 5)/2
First possibility: x = (1 - 5)/2 = - 2 → where: xy = 6 → y = 6/x → y = - 3
Second possibility: x = (1 + 5)/2 = 3 → where: xy = 6 → y = 6/x → y = 2
Second case: xy = (7 - 5)/2 = 1
x + xy - y = 7
x + 1 - y = 7
x - y = 6
y = x - 6 → restart:
xy = 1
x.(x - 6) = 1
x² - 6x = 1
x² - 6x - 1 = 0
Δ = (- 6)² - (4 * - 1) = 40
x = (6 ± √40)/2
x = (6 ± 2√10)/2
x = 3 ± √10
Third possibility: x = 3 + √10 → where: xy = 1 → y = 1/x
y = 1/(3 + √10)
y = (3 - √10)/[(3 + √10).(3 - √10)]
y = (3 - √10)/[9 - 10]
y = - 3 + √10
Fourt possibility: x = 3 - √10 → where: xy = 1 → y = 1/x
y = 1/(3 - √10)
y = (3 + √10)/[(3 - √10).(3 + √10)]
y = (3 + √10)/[9 - 10]
y = - 3 - √10
Summarize (x ; y)
(- 2 ; - 3)
(3 ; 2)
(3 + √10 ; - 3 + √10)
(3 - √10 ; - 3 - √10)

[@E.h.a.b]

x + x y - y = 7 ==> x y+(x-y) = 7
x^2 y - x y^2 = 6 ==> x y (x-y) = 6
Let (x-y) = a & x y = b we get
b+a = 7 -----> [1]
b.a = 6 -----> [2]
(a-b)^2 = (a+b)^2 - 4 ab
(a-b)^2 = 7^2 - 4 * 6 = 25 //From [1] & [2]
a-b = +/- 5 -----> [3]
2 b = 7 -(+/- 5) //From [1]-[3]
**** b = 6 **** OR **** b = 1 ****
(x-y) =a = 7-b = 7-6 = 1 (x-y) =a = 7-b = 7-1 = 6
(x+y)^2 = (x-y)^2 + 4 x y (x+y)^2 = (x-y)^2 + 4 x y
(x+y)^2 = a^2 + 4 (7-a) (x+y)^2 = a^2 + 4 (7-a)
(x+y)^2 = 1^2 + 4(7-1) (x+y)^2 = 6^2 + 4 (7-6)
(x+y)^2 = 25 (x+y)^2 = 40
x+y = +/-5 x+y = +/- 2√10
2 x = 1 +/-5 2 x = 6 +/- 2√10
x = (1 +/- 5)/2 x = (6 +/- 2√10)/2 = (3+/-√10)
x = {3,-2} x = {3+√10, 3-√10}
y = {3,-2} -1 y = {3+√10, 3-√10} -6
y = {2,-3} y = {-3+√10, -3-√10}
Final answer (x,y) = {(3,2), (-2,-3), (3+√10, -3+√10), (3-√10, -3-√10)}