世界上最難的數學問題

1) Two numbers differ by 8. the squares of these two numbers together with their product add up to 19.Find the two numbers.
Let x be the greater number and y be the smaller number
x- y = 8------------- (1)
x2+ y2 + xy = 19 ---(2)
From (2),
x2+ y2 + xy = 19
(x-y)2+3xy= 19 ----- (3)
Sub. (1) into (3),
82 + 3xy = 19
3xy= - 45
xy = -15 ----------- (4)
From (1), x-y = 8, sub. x = y+8 into (4)
(y+8)y = -15
y2 + 8y = -15
y2 + 8y + 15= 0
(y+3)(y+5) = 0
y = -3 or y = -5
when y = -3,
x = -3 + 8 = 5
when y = -5,
x = -5 + 8 = 3
∴ The two numbers are -3 and 5 or -5 and 3.
2) A two number is greater than the product of its gigits by 35 Whwn the digits are reversed ,the new number formed is less than the original one by 9.Find the number.
Let x be the tens digit and y be the units digit. Then the value of the number is (10x+y)
(10x+y) - xy= 35 ----------- (1)
(10x+y) - (10y+x) = 9 ----- (2)
From (2),
10x+y- 10y-x = 9
9x - 9y = 9
x- y = 1
y = x-1 ----------- (3)
sub. (3) into (1),
[10x+(x-1)] - x(x-1)= 35
(11x-1) - x2 + x = 35
12x - 36 - x2 = 0
x2 -12x +36 = 0
(x-6)(x-6) = 0
x = 6 (repeated)
sub x= 6 into (3),
y = 6-1 = 5
∴ The original number is 65.

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1)
Given
x-y=8
x^2 + y^2 +xy=19

Imply
x^2 +(x-8)^2 +x(x-8)=19
3x^2 – 24x +45=0

Solve above equation by Quadratic Equation

x=5 or 3

Imply
x=3 and y=-5 or
x=5 and y=-3

For second question, just too many spelling problems, I cannot understand!