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\(P=\sqrt{3\left(x^2-6x+9\right)+1}+\sqrt{4\left(x^2-\frac{1}{2}x+\frac{1}{16}\right)+\frac{179}{4}}\)
\(P=\sqrt{3\left(x-3\right)^2+1}+\sqrt{4\left(x-\frac{1}{4}\right)^2+\frac{179}{4}}\ge1+\frac{\sqrt{179}}{2}=\frac{2+\sqrt{179}}{2}\)
a) \(A=\sqrt{4x^2+4x+2}=\sqrt{4x^2+4x+1+1}=\sqrt{\left(2x+1\right)^2+1}\)
Vì \(\left(2x+1\right)^2\ge0\forall x\)\(\Rightarrow\left(2x+1\right)^2+1\ge1\forall x\)
\(\Rightarrow A\ge\sqrt{1}=1\)
Dấu " = " xảy ra \(\Leftrightarrow2x+1=0\)\(\Leftrightarrow2x=-1\)\(\Leftrightarrow x=\frac{-1}{2}\)
Vậy \(minA=1\Leftrightarrow x=\frac{-1}{2}\)
b) \(B=\sqrt{2x^2-4x+5+1}=\sqrt{2x^2-4x+2+3+1}=\sqrt{2\left(x^2-2x+1\right)+4}\)
\(=\sqrt{2\left(x-1\right)^2+4}\)
Vì \(\left(x-1\right)^2\ge0\forall x\)\(\Rightarrow2\left(x-1\right)^2\ge0\forall x\)\(\Rightarrow2\left(x-1\right)^2+4\ge4\forall x\)
\(\Rightarrow B\ge\sqrt{4}=2\)
Dấu " = " xảy ra \(\Leftrightarrow x-1=0\)\(\Leftrightarrow x=1\)
Vậy \(minB=2\Leftrightarrow x=1\)
Lời giải:
a. ĐKXĐ: $x\geq 0$
$2\sqrt{2x}-5\sqrt{8x}+7\sqrt{18x}=28$
$\Leftrightarrow 2\sqrt{2x}-10\sqrt{2x}+21\sqrt{2x}=28$
$\Leftrightarrow 13\sqrt{2x}=28$
$\Leftrightarrow \sqrt{2x}=\frac{28}{13}$
$\Leftrightarrow 2x=\frac{784}{169}$
$\Leftrightarrow x=\frac{392}{169}$
b. ĐKXĐ: $x\geq 5$
PT $\Leftrightarrow \sqrt{4}.\sqrt{x-5}+\sqrt{x-5}-\frac{1}{3}.\sqrt{9}.\sqrt{x-5}=4$
$\Leftrightarrow 2\sqrt{x-5}+\sqrt{x-5}-\sqrt{x-5}=4$
$\Leftrightarrow 2\sqrt{x-5}=4$
$\Leftrightarrow \sqrt{x-5}=2$
$\Leftrightarrow x-5=4$
$\Leftrightarrow x=9$ (tm)
c. ĐKXĐ: $x\geq \frac{2}{3}$ hoặc $x< -1$
PT $\Leftrightarrow \frac{3x-2}{x+1}=9$
$\Rightarrow 3x-2=9(x+1)$
$\Leftrightarrow x=\frac{-11}{6}$ (tm)
\(P=-3x^2-4x\sqrt{y}+16x-2y+12\sqrt{y}+1998\)
\(\Leftrightarrow3P=-9x^2-12x\sqrt{y}-4y+16\left(3x+2\sqrt{y}\right)-64-\left(2y-4\sqrt{y}+2\right)+6060\)
\(=-\left(3y+2\sqrt{y}-8\right)^2-2\left(\sqrt{y}-1\right)^2+6060\le6060\)
=> P \(\le2020\)
"=" khi \(\left\{{}\begin{matrix}3x+2\sqrt{y}=8\\\sqrt{y}-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=1\end{matrix}\right.\)
Vậy Min P = 2020 khi x = 2 ; y = 1