Đáp án + Giải thích các bước giải:
d, `2x^2-2x+2`
`= x^2 - x + 1`
`= [x^2-2*x*1/2+(1/2)^2]+3/4`
`= (x-1/2)^2 + 3/4`
Vì `(x-1/2)^2 \ge 0∀x`
`=> (x-1/2)^2 + 3/4 \ge 3/4 > 0 ∀x`
e, `25x^2+10x+2`
`= 25(x^2+10/25x+2/25)`
`= 25[x^2+2*x*1/5+(1/5)^2]+1`
`= 25(x+1/5)^2 + 1`
Vì `25(x+1/5)^2 \ge 0 ∀x`
`=> 25(x+1/5)^2 + 1 \ge 1 > 0 ∀x`
f, `x^2+8x+25`
`= (x^2+2*x*4+4^2) + 9`
`= (x+4)^2 + 9`
Vì `(x+4)^2 \ge 0 ∀x`
`=> (x+4)^2 + 9 \ge 9 > 0 ∀x`
g, `4x^2 - 12x + 31`
`= 4(x^2-3x+31/4)`
`= 4[x^2-2*x*3/2+(3/2)^2]+22`
`= 4(x-3/2)^2 + 22`
Vì `4(x-3/2)^2 \ge 0 ∀x`
`=> 4(x-3/2)^2 + 22 \ge 22 > 0 ∀x`
h, Sửa đề : `x^2 - 2x + y^2 - 4y + 6`
`= (x^2-2x+1) + (y^2-4y+4) + 1`
`= (x-1)^2 + (y-2)^2 + 1`
Vì `(x-1)^2 \ge 0 ∀x`
`(y-2)^2 \ge 0 ∀x`
`=> (x-1)^2 + (y-2)^2 \ge 0 ∀x`
`=> (x-1)^2 + (y-2)^2 + 1 \ge 1 > 0 ∀x`