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7. \(S=9y^2-12\left(x+4\right)y+\left(5x^2+24x+2016\right)\)
\(=9y^2-12\left(x+4\right)y+4\left(x+4\right)^2+\left(x^2+8x+16\right)+1936\)
\(=\left[3y-2\left(x+4\right)\right]^2+\left(x-4\right)^2+1936\ge1936\)
Vậy \(S_{min}=1936\) \(\Leftrightarrow\) \(\hept{\begin{cases}3y-2\left(x+4\right)=0\\x-4=0\end{cases}}\) \(\Leftrightarrow\) \(\hept{\begin{cases}x=4\\y=\frac{16}{3}\end{cases}}\)
8. \(x^2-5x+14-4\sqrt{x+1}=0\) (ĐK: x > = -1).
\(\Leftrightarrow\) \(\left(x+1\right)-4\sqrt{x+1}+4+\left(x^2-6x+9\right)=0\)
\(\Leftrightarrow\) \(\left(\sqrt{x+1}-2\right)^2+\left(x-3\right)^2=0\)
Với mọi x thực ta luôn có: \(\left(\sqrt{x+1}-2\right)^2\ge0\) và \(\left(x-3\right)^2\ge0\)
Suy ra \(\left(\sqrt{x+1}-2\right)^2+\left(x-3\right)^2\ge0\)
Đẳng thức xảy ra \(\Leftrightarrow\) \(\hept{\begin{cases}\left(\sqrt{x+1}-2\right)^2=0\\\left(x-3\right)^2=0\end{cases}}\) \(\Leftrightarrow\) x = 3 (Nhận)
7. \(S=9y^2-12\left(x+4\right)y+\left(5x^2+24x+2016\right)\)
\(=9y^2-12\left(x+4\right)y+4\left(x+4\right)^2+\left(x^2+8x+16\right)+1936\)
\(=\left[3y-2\left(x+4\right)\right]^2+\left(x-4\right)^2+1936\ge1936\)
Vậy \(S_{min}=1936\) \(\Leftrightarrow\) \(\hept{\begin{cases}3y-2\left(x+4\right)=0\\x-4=0\end{cases}}\) \(\Leftrightarrow\) \(\hept{\begin{cases}x=4\\y=\frac{16}{3}\end{cases}}\)
a/ ĐXĐK: ...
\(\Leftrightarrow9x^2-1-x-8x\sqrt{x+1}=0\)
\(\Leftrightarrow x^2-x-1+8x\left(x-\sqrt{x+1}\right)=0\)
\(\Leftrightarrow x^2-x-1+\frac{8x\left(x^2-x-1\right)}{x+\sqrt{x+1}}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2-x-1=0\Rightarrow x=...\\\frac{-8x}{x+\sqrt{x+1}}=1\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow-8x=x+\sqrt{x+1}\)
\(\Leftrightarrow-9x=\sqrt{x+1}\) (\(x\le0\))
\(\Leftrightarrow81x^2-x-1=0\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{1-5\sqrt{13}}{162}\\x=\frac{1+5\sqrt{13}}{162}>0\left(l\right)\end{matrix}\right.\)
d/
\(\Leftrightarrow3x^2+2\left(x^2+x+1\right)-5x\sqrt{x^2+x+1}=0\)
Đặt \(\sqrt{x^2+x+1}=a\)
\(\Leftrightarrow3x^2-5ax+2a^2=0\)
\(\Leftrightarrow\left(x-a\right)\left(3x-2a\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=a\\3x=2a\end{matrix}\right.\) (\(x\ge0\))
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+x+1}=x\\2\sqrt{x^2+x+1}=3x\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+x+1=x^2\\2\left(x^2+x+1\right)=9x^2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-1\left(l\right)\\7x^2-2x-2=0\end{matrix}\right.\) \(\Rightarrow x=\frac{1+\sqrt{15}}{7}\)