10) $D=-30-5x^2+10x$
$D=-(5x^2-10x+30)$
$D=-5(x^2-2x+6)$
$D=-5(x^2-2x+1+5)$
$D=-5[(x-1)^2+5]$
$D=-5(x-1)^2-25$
Do $-5(x-1)^2≤0$ với mọi $x$
$⇒-5(x-1)^2-25≤-25$ với mọi $x$
Hay: $D$ luôn âm.
11) $E=-4x^2-12x-15$
$E=-(4x^2+12x+15)$
$E=-(4x^2+12x+9+6)$
$E=-[(2x+3)^2+6)$
$E=-(2x+3)^2-6$
Do $-(2x+3)^2≤0$ với mọi $x$
$⇒-(2x+3)-6≤-6$ với mọi $x$
Hay: $E$ luôn âm.
12) $F=-\dfrac{1}{2}x^2-2x-6$
$F=-\dfrac{1}{2}(x^2+4x+12)$
$F=-\dfrac{1}{2}(x^2+4x+4+8)$
$F=-\dfrac{1}{2}[(x+2)^2+8]$
$F=-\dfrac{1}{2}(x+2)^2-4$
Do $-\dfrac{1}{2}(x+2)^2≤0$ với mọi $x$
$⇒-\dfrac{1}{2}(x+2)^2-4≤-4$ với mọi $x$
Hay: $F$ luôn âm.
13) $G=-(x-1)(x+2)-5$
$G=(1-x)(x+2)-5$
$G=x+2-x^2-2x-5$
$G=-x^2-x-3$
$G=-(x^2+x+3)$
$G=-\bigg(x^2+x+\dfrac{1}{4}+\dfrac{11}{4}\bigg)$
$G=-\Bigg[\bigg(x+\dfrac{1}{2}\bigg)^2+\dfrac{11}{4}\Bigg]$
$G=-\bigg(x+\dfrac{1}{2}\bigg)^2-\dfrac{11}{4}$
Do $-\bigg(x+\dfrac{1}{2}\bigg)^2≤0$ với mọi $x$
$⇒-\bigg(x+\dfrac{1}{2}\bigg)^2-\dfrac{11}{4}≤-\dfrac{11}{4}$ với mọi $x$
Hay: $G$ luôn âm.
