a) \(A=x^2+2x+2\)

\(=x^2+2x+1+1\)

\(=\left(x+1\right)^2+1>0\forall x\)

b) \(B=4x^2-4x+11\)

\(=4x^2-4x+1+10\)

\(=\left(2x-1\right)^2+10>0\forall x\)

c) \(C=x^2-x+1\)

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

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

d) Ta có: \(D=x^2-2x+y^2+4y+6\)

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

\(=\left(x-1\right)^2+\left(y+2\right)^2+1>0\forall x,y\)

e) Ta có: \(D=x^2-2xy+y^2+x^2-8x+20\)

\(=x^2-2xy+y^2+x^2-8x+16+4\)

\(=\left(x-y\right)^2+\left(x-4\right)^2+4>0\forall x,y\)

\(A=x^2+2x+2=\left(x+1\right)^2+1\ge1>0\left(\forall x\right)\)

\(B=4x^2-4x+11=\left(2x-1\right)^2+10\ge10>0\left(\forall x\right)\)

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

\(D=x^2-2x+y^2+4y+6=x^2-2x+1+y^2+4y+4+1\)

\(=\left(x-1\right)^2+\left(y-2\right)^2+1\ge1>0\)

\(E=x^2-2xy+y^2+x^2-8x+16+4\)

\(=\left(x-y\right)^2+\left(x-4\right)^2+4\ge4>0\)\(\left(\forall x,y\right)\)

a)x^2+2x+3

=x^2+2.x.1+1^2+2

=(x+1)^2+2

         Vì (x+1)^2≥0

   Suy ra:(x+1)^2+2≥(đpcm)

b)-x^2+4x-5

=-(x^2-4x+5)

=-(x^2-2.2x+4)-1

=-(x-2)^2-1

             Vì -(x-2)^2≤0

     Suy ra -(x-2)^2-1≤-1(đpcm)

a) Ta có:

\(x^2-x+1\)

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

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

Mà: \(\left(x-\dfrac{1}{2}\right)^2\ge0\) và \(\dfrac{3}{4}>0\) nên

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

\(\Rightarrow x^2-x+1>0\forall x\)

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 \(x\)

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

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

A=(x+2)^2 +3

B=(x-5)^2 +4

C=4(x+1/2)^2 +4

D=(x-1/2)^2 +19/4

E=2(x-3/4)^2 +95/8

a) x²-6x+10

=x²-6x+9+1

=(x-3)²+1 \(\ge\) 0 (vì (x-3)²\(\ge\)0)

vậy  x^2-6x+10 luôn luôn dương với mọi x

4x-x²-5

=-x²+4x-4-1

=-(x²-4x+4)-1

=-(x-2)²-1\(\le\)-1 ( vì -(x-2)²\(\le\)0 )

vậy 4x-x^2-5 luôn luôn âm với mọi x

A=x^2+x+1 luon luon dương với mọi x

a)x^2+2x+3

=x^2+2.x.1+1^2+2

=(x+1)^2+2

         Vì (x+1)^2≥0

   Suy ra:(x+1)^2+2≥(đpcm)

b)-x^2+4x-5

=-(x^2-4x+5)

=-(x^2-2.2x+4)-1

=-(x-2)^2-1

             Vì -(x-2)^2≤0

     Suy ra -(x-2)^2-1≤-1(đpcm)

a, Ta có: A=x²+2x+3 =x²+2x+1+2

                  = (x+1)²+2>0

b, B= -(x²-4x+5) = -(x²-4x+4)-1

       = -(x-2)²-1<0

Chúc bạn học tốt!

a)x²+2x+3

=x²+2.x.1+1²+2

=(x+1)²+2

         Vì (x+1)²\(\ge0\)

   Suy ra:(x+1)²+2\(\ge2\)(đpcm)

b)-x²+4x-5

=-(x²-4x+5)

=-(x²-2.2x+4)-1

=-(x-2)²-1

             Vì -(x-2)²\(\le0\)

     Suy ra -(x-2)²-1\(\le-1\)(đpcm)

a ) \(x^2+2x+3\)

\(\Rightarrow x^2+2x+1+2\)

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

\(\Rightarrow x^2+2x+3\) luôn dương với mọi nguyên x

b ) \(-x^2+4x-5\)

\(\Rightarrow-\left(x^2-4x+5\right)\)

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

\(\Rightarrow-1-\left(x-2\right)^2\Leftarrow-1\)

\(\Rightarrow-x^2+4x-5\) luôn âm với mọi nguyên x

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

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

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