Question

[Trần Nghiên Hy]

bài 2. Cmr bieur thức luôn âm

6. -2x^2+2x-4

7. -2x^2+6x-8

8. -x^2 +4x-1

9. -2x^2+6x-12

10. 6x^2+15x-21

Answer 1

[Hoàng Thị Ngọc Anh]

6. \(-2x^2+2x-4=-2\left(x^2-x+2\right)\)

\(=-2\left(x^2-2.x.\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{4}+2\right)\)

\(=-2\left[\left(x-\dfrac{1}{2}\right)^2+\dfrac{7}{4}\right]\)

\(=-2\left(x-\dfrac{1}{2}\right)^2-\dfrac{7}{2}\le\dfrac{-7}{2}< 0\)

-> ĐPCM.

7. 8. 9 Tương tự

10. \(6x^2+15x-21\)

\(=6\left(x^2+\dfrac{15}{6}x-\dfrac{7}{2}\right)\)

\(=6\left(x^2+2.x.\dfrac{5}{4}+\dfrac{25}{16}-\dfrac{25}{16}-\dfrac{7}{2}\right)\)

\(=6\left[\left(x+\dfrac{5}{4}\right)^2-\dfrac{81}{16}\right]\)

\(=6\left(x+\dfrac{5}{4}\right)^2-\dfrac{243}{8}\)

\(=\dfrac{243}{8}-6\left(x+\dfrac{5}{4}\right)^2\)

....

Answer 2

[Hoàng Thị Ngọc Mai]

1)

-2x²+2x-4

= -2(x² -x +2)

= -2(x² - 2.\(\dfrac{1}{2}\).x + \(\dfrac{1}{4}\)+\(\dfrac{7}{4}\))

= -2.(x-\(\dfrac{1}{2}\))² - \(\dfrac{7}{2}\) \(\le\) - \(\dfrac{7}{2}\) với \(\forall\) x

=> -2x²+2x-4 luôn âm

=>đpcm

2)

-2x² +6x -8

= -2 (x² -3x + 4)

= -2(x² - 2.\(\dfrac{3}{2}\).x +\(\dfrac{9}{4}\)+\(\dfrac{7}{4}\))

= -2.(x-\(\dfrac{3}{2}\))² - \(\dfrac{7}{2}\) \(\le\) - \(\dfrac{7}{2}\) với \(\forall\) x

=> -2x² +6x -8 luôn âm

=>đpcm

3)

-x² + 4x -1

= - (x² - 4x +1)

= -(x² - 2.2.x + 4 -3)

= -(x - 2)² +3 \(\le\) 3 với \(\forall\) x

=> -x² + 4x -1 có thể không âm

=> sai đề

4)

-2x² +6x -12

= -2(x²- 3x + 6)

= -2(x² - 2.\(\dfrac{3}{2}\).x + \(\dfrac{9}{4}+\dfrac{15}{4}\))

= -2(x-\(\dfrac{3}{2}\))² - \(\dfrac{15}{2}\) \(\le\) - \(\dfrac{15}{2}\) với \(\forall\) x

=> -2x² +6x -12 luôn âm

=>đpcm

5)

6x² +15x - 21

= 6(x² + 2.\(\dfrac{15}{2}\)x + \(\dfrac{225}{4}\)- \(\dfrac{309}{4}\))

= 6.(x-\(\dfrac{15}{2}\))² - \(\dfrac{927}{2}\) \(\ge\) - \(\dfrac{927}{2}\) với \(\forall\) x

=> 6x² +15x - 21 có thể không âm

=> đề sai