Solution: A 2-B 2 = 45 indicates that (a+b) (a-b) = 45 =1* 45 = 3 *15 = 5 * 9.
Because a and b are natural numbers.
So a+b=9, a-b=5 or a+b=45, a-b= 1 or a+b= 15, a-b=3.
The solution is a=23, b=22 or a=9, b=6 or a=7, b=2.
(2) If non-zero real numbers A and B satisfy 4a2+b2=4ab, find the value of A/B..
Solution, because both a and b are non-zero real numbers, then B 2 is also non-zero real numbers.
Divide both sides of the equation by b 2 at the same time
Get (2a/b) 2+1= 4a/b (2a/b-1) 2 = 0.
That is 2a/b= 1 a/b= 1/2.
(3) Let y = (x-1) (x-3) (x-4) (x-6)+10. It is proved that no matter what real number X takes, the value of Y is always greater than 0.
Solution: y = (x-1) (x-6) (x-4) (x-3)+10.
=(x2-7x+6)(x2-7x+ 12)+ 10
Let x2-7x+9 = a.
Then =(a+3)(a-3)+ 10.
=a2-9+ 10
=a2+ 1
=(x2-7x+9)^2+ 1
So no matter what real number X takes, the value of Y is always greater than 0.
(4) Decomposition factor: x2+4xy+4y2-4x-8y+3.
Original formula = (x+2y) 2-4 (x+2y)+3.
=(x+2y-3)(x+2y- 1)