Ta có: \(2x^2+2x+1\)

\(=2\left(x^2+x+\frac{1}{2}\right)\)

\(=2\left(x^2+2\cdot x\cdot\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\right)\)

\(=2\left(x+\frac{1}{2}\right)^2+\frac{1}{2}\)

Ta có: \(\left(x+\frac{1}{2}\right)^2\ge0\forall x\)

\(\Rightarrow2\left(x+\frac{1}{2}\right)^2\ge0\forall x\)

\(\Rightarrow2\left(x+\frac{1}{2}\right)^2+\frac{1}{2}\ge\frac{1}{2}\forall x\)

hay \(2x^2+2x+1>0\forall x\)(đpcm)

Lời giải:

$x^2+x+1=x^2+2.x.\frac{1}{2}+(\frac{1}{2})^2+\frac{3}{4}$

$=(x+\frac{1}{2})^2+\frac{3}{4}$

$\geq 0+\frac{3}{4}$

$> 0$

Ta có đpcm.

Ta có:     \(x^2+2x+5\)

       \(=x^2+2x+1+4\)

       \(=\left(x+1\right)^2+4\)\(>0\)      \(\forall x\)

\(\Rightarrow\)\(x^2+2x+5>0\)  \(\forall x\)

hay BĐT luôn có nghiệm với mọi x

P/S: trình bày sai chỗ nào m.n góp ý mk nhé

Lời giải:

Ta thấy:

$9x^2-6x+2=(9x^2-6x+1)+1$

$=[(3x)^2-2.3x+1^2]+1=(3x-1)^2+1$

Vì $(3x-1)^2\geq 0$ với mọi $x$

$\Rightarrow 9x^2-6x+2=(3x-1)^2+1\geq 1>0$ với mọi $x$

Ta có đpcm.

a/ \(x^2+xy+y^2+1=\left(x^2+xy+\frac{y^2}{4}\right)+\frac{3y^2}{4}+1=\left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}+1>0\)

b/ \(x^2+5y^2+2x-4xy-10y+14\)

\(=\left(x^2-4xy+4y^2\right)+2\left(x-2y\right)+1+\left(y^2-6y+9\right)+4\)

\(=\left(x-2y\right)^2+2\left(x-2y\right)+1+\left(y-3\right)^2+4\)

\(=\left(x-2y+1\right)^2+\left(y-3\right)^2+4>0\)

\(\frac{1+sin^2x}{1-sin^2x}=\frac{cos^2x+sin^2x+sin^2x}{cos^2x+sin^2x-sin^2x}=\frac{cos^2x+2sin^2x}{cos^2x}=1+2\left(\frac{sinx}{cosx}\right)^2=1+2tan^2x\)