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a/ \(\left(x+3\right)\left(3\left(x^2+1\right)^2+2\left(x+3\right)^2\right)=5\left(x^2+1\right)^3\)
\(\Leftrightarrow3\left(x+3\right)\left(x^2+1\right)^2+2\left(x+3\right)^3-5\left(x^2+1\right)^3=0\)
\(\Leftrightarrow3\left(x+3\right)\left(x^2+1\right)^2-3\left(x^2+1\right)^3+2\left(x+3\right)^3-2\left(x^2+1\right)^3=0\)
\(\Leftrightarrow3\left(x^2+1\right)^2\left(-x^2+x+2\right)+2\left(-x^2+x+2\right)\left(\left(x+3\right)^2+\left(x+3\right)\left(x^2+1\right)+\left(x^2+1\right)^2\right)=0\)
\(\Leftrightarrow\left(-x^2+x+2\right)\left[3\left(x^2+1\right)^2+2\left(x+3+\dfrac{x^2+1}{2}\right)^2+\dfrac{3\left(x^2+1\right)^2}{4}\right]=0\)
\(\Leftrightarrow-x^2+x+2=0\) (phần ngoặc phía sau luôn dương)
\(\Rightarrow\left[{}\begin{matrix}x=2\\x=-1\end{matrix}\right.\)
b/ \(3\left(x^2+2x-1\right)^2-2\left(x^2+3x-1\right)^2+5\left(x^2+3x-1-\left(x^2+2x-1\right)\right)^2=0\)
Đặt \(\left\{{}\begin{matrix}a=x^2+2x-1\\b=x^2+3x-1\end{matrix}\right.\)
\(3a^2-2b^2+5\left(b-a\right)^2=0\Leftrightarrow8a^2+3b^2-10ab=0\)
\(\Leftrightarrow\left(4a-3b\right)\left(2a-b\right)=0\Leftrightarrow\left[{}\begin{matrix}4a=3b\\2a=b\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}4\left(x^2+2x-1\right)=3\left(x^2+3x-1\right)\\2\left(x^2+2x-1\right)=x^2+3x-1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x^2-x-1=0\\x^2+x-1=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{1+\sqrt{5}}{2}\\x=\dfrac{1-\sqrt{5}}{2}\\x=\dfrac{-1+\sqrt{5}}{2}\\x=\dfrac{-1-\sqrt{5}}{2}\end{matrix}\right.\)
Đặt a = x - 2 => x - 1 = a + 1; x - 3 = a -1
Khi đó, A = (a+1)4 + (a - 1)4 + 6.(a + 1)2 .(a - 1)2
A = [(a + 1)2 + (a - 1)2]2 + 4.(a + 1)2 .(a - 1)2
= (a2 + 2a + 1 + a2 - 2a + 1)2 + 4.(a2 - 1)2
= (2a2 +2)2 + 4.(a4 - 2a2 + 1)
= 4a4 + 8a2 + 4 + 4a4 - 8a2 + 4 = 8a4 + 8 \(\ge\) 8 với mọi a
=> min A = 8 khi a = 0 <=> x - 2 = 0 <=> x= 2
Ta có:
\(\left|x+\frac{1}{2}\right|\ge0;\left|x+\frac{1}{6}\right|\ge0;....;\left|x+\frac{1}{110}\right|\ge0\)
\(\Rightarrow VT\ge0\Rightarrow VP\ge0\)
\(\Rightarrow x+\frac{1}{2}+x+\frac{1}{6}+...+\frac{1}{110}=11x\)
\(\Rightarrow\left(x+...+x\right)+\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{110}\right)=11x\)
\(\Rightarrow10x+\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{10\cdot11}\right)=11x\)
\(\Rightarrow10x+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{10}-\frac{1}{11}\right)=11x\)
\(\Rightarrow10x+\left(1-\frac{1}{11}\right)=11x\)
\(\Rightarrow10x+\frac{10}{11}=11x\)
\(\Rightarrow x=\frac{10}{11}\)
mk viết vội nên nhầm dòng thứ 4 từ trên xuống, bn sửa 1/3 thành 1/6 nhé
kq vẫn đúng đấy
\(A=\left(x-1\right)\left(x-8\right)\left(x-4\right)\left(x-5\right)+2002\)
\(\Leftrightarrow A=\left(x^2-9x+8\right)\left(x^2-9x+20\right)+2002\)
Đặt \(x^2-9x+14=y\)
\(\Rightarrow A=\left(y-6\right)\left(y+6\right)+2002\)
\(\Leftrightarrow A=y^2-36+2002\)
\(\Leftrightarrow A=y^2+1966\ge1966\)
Dấu "=" xảy ra khi
\(x^2-9x+14=0\)
\(\Leftrightarrow x=2,7\)
\(\left\{{}\begin{matrix}\left(x^2+2x\right)\left(y^2+2y\right)=9\\\left(x^2+2x\right)+\left(y^2+2y\right)=6\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[6-\left(y^2+2y\right)\right]\left(y^2+2y\right)=9\\\left(x^2+2x\right)=6-\left(y^2+2y\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(y^2+2y\right)^2-6\left(y^2+2y\right)+9=0\\\left(x^2+2x\right)=6-\left(y^2+2y\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(y^2+2y-3\right)^2=0\\\left(x^2+2x\right)=6-\left(y^2+2y\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y^2+2y-3=0\\\left(x^2+2x\right)=6-\left(y^2+2y\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}y=1\\y=-3\end{matrix}\right.\\\left(x^2+2x\right)=6-\left(y^2+2y\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}y=1\\x^2+2x=6-y^2-2y\end{matrix}\right.\\\left\{{}\begin{matrix}y=-3\\x^2+2x=6-y^2-2y\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}y=1\\x^2+2x-3=0\end{matrix}\right.\\\left\{{}\begin{matrix}y=-3\\x^2+2x-3=0\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}y=1\\\left[{}\begin{matrix}x=1\\x=-3\end{matrix}\right.\end{matrix}\right.\\\left\{{}\begin{matrix}y=-3\\\left[{}\begin{matrix}x=1\\x=-3\end{matrix}\right.\end{matrix}\right.\end{matrix}\right.\)
vậy \(S=\left\{\left(1;1\right),\left(-3;1\right),\left(1;-3\right),\left(-3;-3\right)\right\}\)
ko viết lại đề
=> \(2x\left\{156-6\left[54-2.15\right]\right\}=134-86\)
=> \(2x\left\{156-6\left[54-30\right]\right\}=48\)
=> \(2x\left\{156-6.24\right\}=48\)
=> \(2x\left\{156-144\right\}=48\)
=> \(2x.12=48\)
=> \(24x=48\)
=> \(x=48:24=2\)
Vậy \(x\in\left\{2\right\}\)