Inequality高手～～

another method

8-3x-x^2≦M
0≦x^2+3x+(M-8)

therefore the quadratic equation x^2+3x+(M-8)=0 has no real root or equal root
such that, discriminant of x^2+3x+(M-8) is <0 or =0

Discriminant of x^2+3x+(M-8)
= 3^2 - 4(1)(M-8)
= 9-4M+32
= -4M+41 < or = 0
M > or = 41/4

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if ax^2+bx+c=0
Discriminant = b^2-4ac
if Discriminant > 0, the quadratic equation has 2 roots
if Discriminant = 0, the quadratic equation has 1 root
if Discriminant < 0, the quadratic equation has no real root (2 imaginery roots)

By completing square,
8 - 3x - x^2
= -(x^2 + 3x) + 8
= -[x^2 + 3x + (3/2)^2] + (3/2)^2 + 8
= -(x + 3/2)^2 + 9/4 + 8
= -(x + 3/2)^2 + 41/4

Since -(x + 3/2)^2 ≦ 0 for all real values of x,
Therefore, -(x + 3/2)^2 + 41/4 ≦ 41/4
That is, 8 - 3x - x^2 ≦ 41/4

So, 41/4 ≦ M