a) \(x^2+x+1=x^2+x+\frac{1}{4}+\frac{3}{4}=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}>0\forall x\)

c) \(C=4x-10-x^2=-\left(x^2-4x+10\right)\)

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

\(=-\left(x^2-4x+4+6\right)=-\left[\left(x-2\right)^2\right]-6\le-6< 0\forall x\)

a: \(A=x^2-x+1\)

\(=x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}\)

\(=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\forall x\)

b: \(B=-x^2+4x-17\)

\(=-\left(x^2-4x+17\right)\)

\(=-\left(x^2-4x+4+13\right)\)

\(=-\left(x-2\right)^2-13< 0\forall x\)

a) \(A=x^2-x+1=\left(x^2-x+\dfrac{1}{4}\right)+\dfrac{3}{4}=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}>0\)

b) \(4x-17-x^2=-\left(x^2-4x+4\right)-13=-\left(x-2\right)^2-13\le-13< 0\)

a) A = \(x^2-x+1\) 

        = \(x^2\) - 2.\(x\).\(\dfrac{1}{2}\) + \(\left(\dfrac{1}{2}\right)^2\) + \(\dfrac{3}{4}\)

         = \(\left(x-\dfrac{1}{2}\right)^2\) + \(\dfrac{3}{4}\)

Với mọi \(x\) ta có:

            \(\left(x-\dfrac{1}{2}\right)^2\) ≥ 0

        ➩\(\left(x-\dfrac{1}{2}\right)^2\) + \(\dfrac{3}{4}\) > \(\dfrac{3}{4}\)

        ➩\(\left(x-\dfrac{1}{2}\right)^2\)  + \(\dfrac{3}{4}\) > 0

        ➩\(x^2-x+1\) > 0

         ➩ A > 0

Vậy biểu thức A = \(x^2-x+1\) luôn dương với mọi \(x\)

a) \(A=x^2+2x+3=x^2+2x+1+2\)

\(=\left(x+1\right)^2+2\ge2\)

Vậy A luôn dương với mọi x

b) \(B=-x^2+4x-5=-\left(x^2-4x+5\right)\)

\(=-\left(x^2-4x+2^2\right)-1\)

\(=-\left(x-2\right)^2-1\le-1\)

Vậy B luôn âm với mọi x

a)\(x^2+2x+3=\left(x^2+2x+1\right)+2=\left(x+1\right)^2+2\ge2\)

Vậy x² +2x+3 luôn dương.

b)\(-x^2+4x-5=-\left(x^2-4x+5\right)=-\left(x^2-4x+4+1\right)=-\left[\left(x-2\right)^2+1\right]\le-1\)

Vậy -x² +4x-5 luôn luôn âm.

sai đề rồi x=-2;y=5 biểu thức dương

Ta có A = -x² + 4x - 6 - y² - 2y 

= -(x² - 4x + 4) - (y² + 2y + 1) - 1

= -(x - 2)² - (y + 1)² - 1 \(\le-1< 0\)

=> A < 0 với mọi x ; y

A = -x² + 4x - 6 - y² - 2y 

= -( x² - 4x + 4 ) - ( y² + 2y + 1 ) - 1

= -( x - 2 )² - ( y - 1 )² - 1 ≤ -1 < 0 ∀ x, y

=> đpcm

\(A=-x^2+4x-6-y^2-2y\)

\(=-x^2+4x-4-y^2-2y-1-1\)

\(=-\left(x^2-4x+4\right)-\left(y^2+2y+1\right)-1\)

\(=-\left(x-2\right)^2-\left(y+1\right)^2-1\)

\(=-\left[\left(x-2\right)^2+\left(y+1\right)^2+1\right]\)

mà \(\left(x-2\right)^2\ge0\forall x\)

      \(\left(y+1\right)^2\ge0\forall y\)

\(\Rightarrow\left(x-2\right)^2+\left(y+1\right)^2\ge0\forall x,y\)

\(\Rightarrow\left(x-2\right)^2+\left(y+1\right)^2+1\ge1\)

\(\Rightarrow\left(x-2\right)^2+\left(y+1\right)^2+1>0\)

\(\Rightarrow-\left[\left(x-2\right)^2+\left(y+1\right)^2+1\right]< 0\)

\(\Rightarrow A< 0\)

Vậy A luôn có giá trị âm với mọi x,y

Bài 1

\(A=x^2-6x+15=x^2-2.3.x+9+6=\left(x-3\right)^2+6>0\forall x\)

\(B=4x^2+4x+7=\left(2x\right)^2+2.2.x+1+6=\left(2x+1\right)^2+6>0\forall x\)

Bài 2

\(A=-9x^2+6x-2021=-\left(9x^2-6x+2021\right)=-\left[\left(3x-1\right)^2+2020\right]=-\left(3x-1\right)^2-2020< 0\forall x\)

2. Ta có: P = 2x² + y² - 4x - 4y + 10

P = 2(x² - 2x + 1) + (y² - 4y + 4) + 4

P = 2(x - 1)² + (y - 2)² + 4 \(\ge\)4 \(\forall\)x;y

=> P luôn dương với mọi biến x;y

3 Ta có:

(2n + 1)(n² - 3n - 1) - 2n³ + 1

= 2n³ - 6n² - 2n + n² - 3n - 1 - 2n³ + 1

= -5n² - 5n = -5n(n + 1) \(⋮\)5 \(\forall\)n \(\in\)Z

1×2=2

Ta có : C = 4x² + 4y² - 8x + 4y + 427

=> C = (4x² - 8x + 4) + (4y² + 4y + 1) + 422

=> C = (2x - 2)² + (2y + 1)² + 422

Mà \(\left(2x-2\right)^2\ge0\forall x\)

       \(\left(2y+1\right)^2\ge0\forall x\)

Nên C = (2x - 2)² + (2y + 1)² + 422  \(\ge422\forall x\)

Suy ra : C = (2x - 2)² + (2y + 1)² + 422 \(>0\forall x\)

Vậy C luôn luôn dương (đpcm)

\(4\left(x^2-x+\dfrac{1}{4}\right)-1+3=4\left(x-\dfrac{1}{2}\right)^2+2\)

mà \(4\left(x-\dfrac{1}{2}\right)^2\ge0\) với mọi x

\(\Rightarrow4\left(x-\dfrac{1}{2}\right)^2+2>0\) với mọi x

\(\Rightarrow dpcm\)

\(A=4x^2-4x+3=4\left(x^2-x+\dfrac{1}{4}\right)-1+3=4\left(x-\dfrac{1}{2}\right)^2+2\)

mà \(4\left(x-\dfrac{1}{2}\right)^2\ge0\) với mọi x