Question

[Minh Hoang Hai]

CM răng đa thức sau ko có nghiệm : \(P\left(x\right)=2x^2+2x+\frac{5}{4}\)

Answer 1

[ngonhuminh]

\(P\left(x\right)=2x^2+2x+\frac{5}{4}=\left(2x^2+x\right)+\left(x+\frac{1}{2}\right)+\left(\frac{5}{4}-\frac{1}{2}\right)\)

\(P\left(x\right)=x\left(2x+1\right)+\frac{1}{2}\left(2x+1\right)+\frac{3}{2}\)

\(P\left(x\right)=\left(2x+1\right)\left(x+\frac{1}{2}\right)+\frac{3}{2}=\left(2x+1\right)\frac{\left(2x+1\right)}{2}+\frac{3}{2}\)

\(p\left(x\right)=\frac{\left(2x+1\right)^2}{2}+\frac{3}{2}=\frac{\left(2x+1\right)^2+3}{2}\)

\(\left(2x+1\right)^2\ge0\Rightarrow\left(2x+1\right)^2+3\ge3\Rightarrow p\left(x\right)>0\forall x\Rightarrow dpcm\)