Ta có: \(\left(a-b\right)^2\ge0,\forall ab\)

         \(\Leftrightarrow a^2-2ab+b^2\ge0\)

           \(\Leftrightarrow a^2+b^2\ge2ab\)

        \(\Leftrightarrow a^2+2ab+b^2\ge4ab\)

         \(\Leftrightarrow\left(a+b\right)^2\ge4ab\left(1\right)\)

Lại có:  \(a^2+b^2\ge2ab\)

         \(\Leftrightarrow2a^2+2b^2\ge a^2+2ab+b^2\)

         \(\Leftrightarrow2\left(a^2+b^2\right)\ge\left(a+b\right)^2\left(2\right)\)

Từ (1) và (2) suy ra ĐPCM

khó quá

c)\(\frac{a^2}{b^2}+\frac{b^2}{a^2}+4\ge3\cdot\left(\frac{a}{b}+\frac{b}{a}\right)\)

Thế : \(\frac{\left(a-b\right)^2\left(a^2-ab+b^2\right)}{a^2b^2}\ge0\)

\(\Leftrightarrow\frac{\left(b-a\right)^2\left(a^2-ab+b^2\right)}{a^2b^2}\ge0\)

\(\Leftrightarrow\frac{a^4+4a^2b^2+b^4}{a^2b^2}\ge\frac{3\left(a^2+b^2\right)}{ab}\)

\(\Leftrightarrow\frac{a^2}{b^2}+\frac{b^2}{a^2}+4\ge\frac{3a}{b}+\frac{3b}{a}\)

\(\Leftrightarrow\left(a-b\right)^2\ge0\)(luôn đúng)

\(\Rightarrow\frac{a^2}{b^2}+\frac{b^2}{a^2}+4>=3\cdot\left(\frac{a}{b}+\frac{b}{a}\right)\)

Mấy câu khác mình đang suy nghĩ nhé

\(\left(x^2+6x+5\right)\left(x+4\right)\left(x+2\right)=40\)

\(\Leftrightarrow\left(x^2+6x+5\right)\left(x^2+6x+8\right)=40\)

Đặt \(x^2+6x+5=t\) ,ta có:

\(t\left(t+3\right)=40\)

\(\Leftrightarrow t^2+3t-40=0\)

\(\Leftrightarrow t^2+8t-5t-40=0\)

\(\Leftrightarrow\left(t+8\right)\left(t-5\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=-8\\t=5\end{matrix}\right.\)

Với t = -8

\(x^2+6x+5=-8\)

\(\Leftrightarrow x^2+6x+13=0\) ( vô lý vì \(x^2+6x+13>0\forall x\) )

Với t = 5

\(x^2+6x+5=5\)

\(\Leftrightarrow x^2+6x=0\)

\(\Leftrightarrow x\left(x+6\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-6\end{matrix}\right.\)

Vậy ............................

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