Toán Lớp 8: Chứng minh $\frac{x^2+x+1}{2}$ lớn hơn 0 (với mọi x khác -1)
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Toán Lớp 8: Chứng minh $\frac{x^2+x+1}{2}$ lớn hơn 0 (với mọi x khác -1)
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\quad \dfrac{x^2 + x + 1}{2}\\
= \dfrac{x^2 + 2\cdot x\cdot \dfrac12 + \dfrac14 + \dfrac34}{2}\\
= \dfrac{\left(x + \dfrac12\right)^2 + \dfrac34}{2}\\
\text{Ta có:}\\
\quad \left(x + \dfrac12\right)^2\geqslant 0\quad \forall x\\
\Leftrightarrow \left(x + \dfrac12\right)^2+\dfrac{3}{4} >0\\
\Leftrightarrow \dfrac{\left(x + \dfrac12\right)^2 + \dfrac34}{2}>0\\
\text{Vậy}\ \dfrac{x^2 + x + 1}{2}>0\quad \forall x
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