Question

1.用配方法证明2x的方减x加1的值恒大于0 2已知关于x的方程x的方减6x减4n的方减32等于0的根是整数求整数n值

Answer 1

1.2x^2 - x + 1 = 2(x^2 - 2 * 1/4 * x + 1/16) + 7/8

= 2(x-1/4)^2 + 7/8 >= 7/8 > 0

2.x^2 -6x - 4n^2 - 32 = 0

x^2 - 6x + 9 - 4n^2 = 41

(x-3)^2 - (2n)^2 = 41

(x - 3 + 2n)*(x-3-2n) = 41

因为 41 是素数,所以共有 4种可能：

a)

x - 3 + 2n = 41

x - 3 - 2n = 1

解得：x = 24 ,n = 10

b)

x - 3 + 2n = 1

x - 3 - 2n = 41

解得：x = 24 ,n = -10

c)

x - 3 + 2n = -41

x - 3 - 2n = -1

解得：x = -18 ,n = -10

d)

x - 3 + 2n = -1

x - 3 - 2n = -41

解得：x = -18 ,n = 10

所以 n = 10 或 -10