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Đk: `1 <=x <=7`.
Đặt `sqrt(7-x) = a, sqrt(x-1) = b`.
Phương trình trở thành: `b^2+1 + 2a = 2b + ab + 1`.
`<=> b^2 + 2a = 2b + ab.`
`<=> b(b-2) = a(b-2)`
`<=> (b-a)(b-2) = 0`
`<=> a =b` hoặc `b = 2.`
`@ a = b => 7 - x = x - 1`
`<=> 8 = 2x <=> x = 4`.
`@ b = 2 => sqrt(x-1) = 2`
`<=> x - 1 = 4`
`<=> x = 5`.
Vậy `x = 4` hoặc `x = 5`.
\(\text{ĐKXĐ:}1\le x\le7\)
PT đã cho tương đương với:
\(x-1-2\sqrt{x-1}+2\sqrt{7-x}-\sqrt{x-1}.\sqrt{7-x}=0\)
\(\Leftrightarrow\sqrt{x-1}\left(\sqrt{x-1}-2\right)-\sqrt{7-x}\left(\sqrt{x-1}-2\right)=0\)
\(\Leftrightarrow\left(\sqrt{x-1}-\sqrt{7-x}\right)\left(\sqrt{x-1}-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x-1}=\sqrt{7-x}\\\sqrt{x-1}=2\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x-1=7-x\\x-1=4\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=4\left(tm\right)\\x=5\left(tm\right)\end{matrix}\right.\)
Vậy pt có tập nghiệm \(S=\left\{4;5\right\}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
1. ĐKXĐ: $x\geq \frac{-3}{5}$
PT $\Leftrightarrow 5x+3=3-\sqrt{2}$
$\Leftrightarrow x=\frac{-\sqrt{2}}{5}$
2. ĐKXĐ: $x\geq \sqrt{7}$
PT $\Leftrightarrow (\sqrt{x}-7)(\sqrt{x}+7)=4$
$\Leftrightarrow x-49=4$
$\Leftrightarrow x=53$ (thỏa mãn)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt \(t=\sqrt{x^2+4\sqrt{5}}\to t>0.\) Phương trình trở thành \(\frac{\left(2t^2-7\right)^2-161}{4}=\left(34-3t^2\right)t\Leftrightarrow\left(2t^2-7\right)^2-161=4t\left(34-3t^2\right)\)
\(\Leftrightarrow\left(t^2-2t-4\right)\left(t^2+5t+7\right)=0\Leftrightarrow t^2-2t=4\Leftrightarrow t=1+\sqrt{5}.\) (Vì t>0)
Vậy ta được \(x^2+4\sqrt{5}=\left(1+\sqrt{5}\right)^2\Leftrightarrow x^2=\left(\sqrt{5}-1\right)^2\Leftrightarrow x=\pm\left(\sqrt{5}-1\right).\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a) Áp dụng bđt AM-GM có:
\(\sqrt[3]{\left(9-x\right).8.8}\le\dfrac{9-x+8+8}{3}=\dfrac{25-x}{3}\)\(\Leftrightarrow\sqrt[3]{9-x}\le\dfrac{25-x}{12}\)
\(\sqrt[3]{\left(7+x\right).8.8}\le\dfrac{7+x+8+8}{3}=\dfrac{23+x}{3}\)\(\Leftrightarrow\sqrt[3]{7+x}\le\dfrac{23+x}{12}\)
Cộng vế với vế \(\Rightarrow\sqrt[3]{9-x}+\sqrt[3]{7+x}\le4\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}9-x=8\\7+x=8\end{matrix}\right.\)\(\Rightarrow x=1\)
Vậy...
b)Đk:\(x\ge2\)
Pt \(\Leftrightarrow\left(x-1\right)^2.\left(x^2-4\right)=\left(x-2\right)^2.\left(x^2-1\right)\)
\(\Leftrightarrow\left(x-1\right)^2\left(x-2\right)\left(x+2\right)=\left(x-2\right)^2\left(x+1\right)\left(x-1\right)\)
Do \(x\ge2\Rightarrow x-1>0\)
Chia cả hai vế của pt cho x-1 ta được:
\(\left(x-1\right)\left(x-2\right)\left(x+2\right)=\left(x-2\right)^2\left(x+1\right)\)
\(\Leftrightarrow\left(x-2\right)\left[\left(x-1\right)\left(x+2\right)-\left(x-2\right)\left(x-1\right)\right]=0\)
\(\Leftrightarrow\left(x-2\right)\left[x^2+x-2-x^2+3x-2\right]=0\)
\(\Leftrightarrow\left(x-2\right)\left(4x-4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\left(tm\right)\\x=1\left(ktm\right)\end{matrix}\right.\)
Vậy S={2}
c)Đk:\(\left\{{}\begin{matrix}9-x^2\ge0\\x^2-1\ge0\\x-3\ge0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}-3\le x\le3\\\left[{}\begin{matrix}x\ge1\\x\le-1\end{matrix}\right.\\x\ge3\end{matrix}\right.\)\(\Rightarrow x=3\)
Thay x=3 vào pt thấy thỏa mãn
Vậy S={3}
a) Quên mất, ko áp dụng đc AM-GM, xin lỗi
Pt \(\Leftrightarrow\sqrt[3]{9-x}-2=2-\sqrt[3]{7+x}\)
\(\Leftrightarrow\dfrac{9-x-8}{\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4}=\dfrac{8-\left(7-x\right)}{4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}}\)
\(\Leftrightarrow\dfrac{1-x}{\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4}=\dfrac{1-x}{4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-1=0\\\dfrac{1}{\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4}=\dfrac{1}{4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4=4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}\left(1\right)\end{matrix}\right.\)
Từ (1) \(\Leftrightarrow\sqrt[3]{\left(9-x\right)^2}-\sqrt[3]{\left(7+x\right)^2}+2\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)=0\)
\(\Leftrightarrow\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)\left(\sqrt[3]{9-x}+\sqrt[3]{7+x}\right)+2\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)=0\)
\(\Leftrightarrow\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right).4+2\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)=0\)
\(\Leftrightarrow\sqrt[3]{9-x}-\sqrt[3]{7+x}=0\)
\(\Leftrightarrow\sqrt[3]{9-x}=\sqrt[3]{7+x}\)\(\Leftrightarrow9-x=7+x\)
\(\Leftrightarrow x=1\)
Vậy S={1}
![](https://rs.olm.vn/images/avt/0.png?1311)
4) Ta có: \(\left(x+3\right)\cdot\sqrt{10-x^2}=x^2-x-12\)
\(\Leftrightarrow\left(x+3\right)\cdot\sqrt{10-x^2}-\left(x-4\right)\left(x+3\right)=0\)
\(\Leftrightarrow\left(x+3\right)\left(\sqrt{10-x^2}-x+4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+3=0\\\sqrt{10-x^2}=x-4\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-3\\10-x^2=x^2-8x+16\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-3\\x^2-8x+16-10+x^2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-3\\2x^2-8x+6=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-3\\2\left(x^2-4x+3\right)=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-3\\\left(x-1\right)\left(x-3\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-3\\x=1\\x=3\end{matrix}\right.\)
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<=> \(\sqrt{\left(x-2\right)-2\sqrt{x-2}+1}+\sqrt{\left(x-2\right)+6\sqrt{x-2}+9}=2\) (đkxđ x\(\ge2\))
<=> l\(\sqrt{x-2}-1\)l +\(\sqrt{x-2}+3\)=2
<=> l\(1-\sqrt{x-2}\)l +\(\sqrt{x-2}+3\)=2
dễ thấy VT \(\ge\)2 =VP (vì l\(1-\sqrt{x-2}\)l \(\ge\)\(1-\sqrt{x-2}\))
=> VT = VP <=> l\(1-\sqrt{x-2}\)l = \(1-\sqrt{x-2}\)
<=> \(1-\sqrt{x-2}\ge0\)
<=> \(\sqrt{x-2}\le1\)
=> x-2 \(\le\)1
<=> x\(\le\)3
kh với đkxđ => \(2\le x\le3\)
\(\Leftrightarrow\sqrt{x-1}-1+\sqrt{x+2}-2=\sqrt{x+34}-6+3-\sqrt{x+7}\)
ĐK: x>=1
\(pt\Leftrightarrow\left(\sqrt{x-1}-1\right)+\left(\sqrt{x+2}-2\right)+\left(\sqrt{x+7}-3\right)-\left(\sqrt{x+34}-6\right)=0\)
\(\Leftrightarrow\frac{x-2}{\sqrt{x-1}+1}+\frac{x-2}{\sqrt{x+2}+2}+\frac{x-2}{\sqrt{x+7}+3}-\frac{x-2}{\sqrt{x+34}+6}=0\)
\(\Leftrightarrow\left(x-2\right)\left(\frac{1}{\sqrt{x-1}+1}+\frac{1}{\sqrt{x+2}+2}+\frac{1}{\sqrt{x+7}+3}-\frac{1}{\sqrt{x+34}+6}\right)=0\left(1\right)\)
Vì trong ngoặc lớn hơn 0 mọi x>=1
phương trình (1) <=> x-2=0
<=>x=2 ( thỏa mãn)