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\(=>x^3=(\sqrt[3]{2\left(\sqrt{3}+1\right)}-\sqrt[3]{2\left(\sqrt{3}-1\right)})^3\)
\(x^3=2\left(\sqrt{3}+1\right)-3.\left[\sqrt[3]{2\left(\sqrt{3}+1\right)}\right]^2.\left[\sqrt[3]{2\left(\sqrt{3}-1\right)}\right]\)
+\(3\left[\sqrt[3]{2\left(\sqrt{3}-1\right)}\right]^2\left[\sqrt[3]{2\left(\sqrt{3}+1\right)}\right]-2\left(\sqrt{3}-1\right)\)
\(x^3=\)
\(4-3\left[\sqrt[3]{2\left(\sqrt{3}+1\right)}\right]\left[\sqrt[3]{2\left(\sqrt{3}-1\right)}\right]\left[\sqrt[3]{2\left(\sqrt{3}+1\right)}-\sqrt[3]{2\left(\sqrt{3}-1\right)}\right]\)
\(x^3=4-3.\left[\sqrt[3]{4\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\right].\)\(x\)
\(x^3=4-3\left[\sqrt[3]{4\left(3-1\right)}\right].x\)
\(x^3=4-3.2x\)
\(x^3=4-6x\)
thay \(x^3=4-6x\) vào A=>\(A=\left(4-6x+6x-5\right)^{2009}=\left(-1\right)^{2009}=-1\)
1. \(\sqrt{\left(x+3\right)\left(x+7\right)}=3\sqrt{x+3}+2\sqrt{x+7}-6\)
\(\Leftrightarrow\sqrt{\left(x+3\right)\left(x+7\right)}-3\sqrt{x+3}-2\sqrt{x+7}+6=0\)
\(\Leftrightarrow\sqrt{x+3}\left(\sqrt{x+7}-3\right)-2\left(\sqrt{x+7}-3\right)=0\)
\(\Leftrightarrow\left(\sqrt{x+7}-3\right)\left(\sqrt{x+3}-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x+7}-3=0\\\sqrt{x+3}-2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\sqrt{x+7}=3\\\sqrt{x+3}=2\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2\\x=1\end{matrix}\right.\)
Vậy...
2. \(2x^2+2x+1=\sqrt{4x+1}\)
\(\Leftrightarrow2x^2+2x+1-\sqrt{4x+1}=0\)
\(\Leftrightarrow4x^2+4x+2-2\sqrt{4x+1}=0\)
\(\Leftrightarrow4x+1-2\sqrt{4x+1}+1+4x^2=0\)
\(\Leftrightarrow\left(\sqrt{4x+1}-1\right)^2+4x^2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{4x+1}=1\\2x=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}4x+1=1\\x=0\end{matrix}\right.\)\(\Leftrightarrow x=0\)
Vậy...
3. \(\sqrt{x-2\sqrt{x-1}}+\sqrt{x+2\sqrt{x-1}}=\frac{x+3}{2}\)
\(\Leftrightarrow\sqrt{x-1-2\sqrt{x-1}+1}+\sqrt{x-1+2\sqrt{x-1}+1}=\frac{x+3}{2}\)
\(\Leftrightarrow\sqrt{\left(\sqrt{x-1}-1\right)^2}+\sqrt{\left(\sqrt{x-1}+1\right)^2}=\frac{x+3}{2}\)
\(\Leftrightarrow\left|\sqrt{x-1}-1\right|+\sqrt{x-1}+1=\frac{x+3}{2}\)
Đặt \(\sqrt{x-1}=a\)
\(\Leftrightarrow x-1=a^2\Leftrightarrow x+3=a^2+4\)
\(pt\Leftrightarrow\left|a-1\right|+a+1=\frac{a^2+4}{2}\)
+) Xét \(a\le1\Leftrightarrow a-1\le0\Leftrightarrow1\le x\le2\)
\(pt\Leftrightarrow1-a+a+1=\frac{a^2+4}{2}\)
\(\Leftrightarrow2=\frac{a^2+4}{2}\)
\(\Leftrightarrow a^2+4=4\)
\(\Leftrightarrow a=0\)
\(\Leftrightarrow\sqrt{x-1}=0\)
\(\Leftrightarrow x=1\) ( thỏa )
+) Xét \(a\ge1\Leftrightarrow a-1\ge0\Leftrightarrow x>2\)
\(pt\Leftrightarrow a-1+a+1=\frac{a^2+3}{2}\)
\(\Leftrightarrow2a=\frac{a^2+3}{2}\)
\(\Leftrightarrow a^2+3=4a\)
\(\Leftrightarrow a^2-4a+3=0\)
\(\Leftrightarrow\left(a-1\right)\left(a-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=1\\a=3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2\left(loai\right)\\x=10\left(thoa\right)\end{matrix}\right.\)
Vậy...
b, \(\sqrt[3]{24+x}+\sqrt{12-x}=6\) (đk \(-24\le x\le12\)) (*)
Đặt \(\sqrt[3]{24+x}=a\) , \(\sqrt{12-x}=b\left(b\ge0\right)\)
Có \(a^3+b^2=24+x+12-x=36\)(1)
a+b=6 => b=6-a
Thay b=6-a vào (1) có:
\(a^3+\left(6-a\right)^2=36\)
<=> \(a^3+a^2-12a+36=36\)
<=> \(a^3+a^2-12a=0\)
<=> \(a\left(a^2+a-12\right)=0\)
<=> \(a\left(a^2-3a+4a-12\right)=0\)
<=> \(a\left(a+4\right)\left(a-3\right)=0\)
=>\(\left[{}\begin{matrix}a=0\\a=-4\\a=3\end{matrix}\right.\)<=> \(\left[{}\begin{matrix}24+x=0\\24+x=-4^3=-64\\24+x=3^3=27\end{matrix}\right.\)<=>\(\left[{}\begin{matrix}x=-24\\x=-88\\x=3\end{matrix}\right.\)(tm pt(*))
Vậy pt (*) có tập nghiệm \(S=\left\{-24,-88,3\right\}\)
Lời giải:
Đặt $\sqrt[3]{x+1}=a;\sqrt[3]{x-1}=b$ thì pt trở thành:
\(\left\{\begin{matrix} a^2+b^2+ab=1\\ a^3-b^3=2\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a^2+ab+b^2=1\\ (a-b)(a^2+ab+b^2)=2\end{matrix}\right.\)
\(\Leftrightarrow \left\{\begin{matrix} a^2+ab+b^2=1\\ a-b=2\end{matrix}\right.\)
\(\Rightarrow \left\{\begin{matrix} (a-b)^2+3ab=1\\ a-b=2\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} a(-b)=1\\ a+(-b)=2\end{matrix}\right.\)
Theo đl Viet đảo thì $a,-b$ là nghiệm của pt $X^2-2X+1=0$
$\Rightarrow a=-b=1$
$\Leftrightarrow \sqrt[3]{x+1}=1; \sqrt[3]{x-1}=-1$
$\Rightarrow x=0$
Vậy.........