ĐKXĐ : \(x\ge0;y\ge1\)

\(x+y+12=4\sqrt{x}+6\sqrt{y-1}\)

\(\Leftrightarrow x-4\sqrt{x}+4+y-1-6\sqrt{y-1}+9=0\)

\(\Leftrightarrow\left(\sqrt{x}-2\right)^2+\left(\sqrt{y-1}-3\right)^2=0\)

\(\Leftrightarrow\hept{\begin{cases}\sqrt{x}-2=0\\\sqrt{y-1}-3=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=4\\y=10\end{cases}}}\)

Dat \(a=\sqrt[3]{65+x},b=\sqrt[3]{65-x}\)

Bien doi PT thanh \(a^2+4b^2=5ab\)

\(\Leftrightarrow a^2-5ab+4b^2=0\)

\(\Leftrightarrow\left(a^2-ab\right)-\left(4ab-4b^2\right)=0\)

\(\Leftrightarrow a\left(a-b\right)-4b\left(a-b\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(a-4b\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}a=b\left(1\right)\\a=4b\left(2\right)\end{cases}}\)

\(\left(1\right)\Leftrightarrow\sqrt[3]{65+x}=\sqrt[3]{65-x}\)

\(\Leftrightarrow65+x=65-x\)

\(\Leftrightarrow x=0\left(n\right)\)

\(\left(2\right)\Leftrightarrow\sqrt[3]{65+x}=4\sqrt[3]{65-x}\)

\(\Leftrightarrow65+x=64.65-64x\)

\(\Leftrightarrow65x=64.65-65\)

\(\Leftrightarrow x=63\left(n\right)\)

Vay nghiem cua PT la \(x=0,x=63\)

\(a,9-4\sqrt{x}=1\Rightarrow-4\sqrt{x}=-8\)

\(\Rightarrow\sqrt{x}=2\Leftrightarrow x=4\)

\(b,\sqrt{\frac{x}{5}}=4\Rightarrow\frac{x}{5}=16\)

\(\Rightarrow x=5.16=80\)

\(c,\sqrt{7x}< 9\Leftrightarrow7x< 81\)

\(\Rightarrow x< \frac{81}{7}\)Và \(x\ge0\)

\(\Rightarrow0\le x< \frac{81}{7}\)

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nâng cao và phát triên lớp 6 nha bạn

Ta chứng minh BĐT sau : \(\frac{1}{\sqrt{n}}=\frac{2}{2\sqrt{n}}< \frac{2}{\sqrt{n-1}+\sqrt{n}}=2\left(\sqrt{n}-\sqrt{n-1}\right)\)

Áp dụng BĐT trên, ta có :

\(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}< 2\left(\sqrt{1}-\sqrt{0}+\sqrt{2}-\sqrt{1}+...+\sqrt{n}-\sqrt{n-1}\right)=2\sqrt{n}\)

Câu hỏi của Tăng Thiện Đạt - Toán lớp 8 - Học toán với OnlineMath