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ĐKXĐ: \(x\ge-1\)
\(\Leftrightarrow\sqrt{\left(\sqrt{x+1}+1\right)^2}+\sqrt{\left(\sqrt{x+1}-3\right)^2}=2\sqrt{\left(\sqrt{x+1}-1\right)^2}\)
\(\Leftrightarrow\left|\sqrt{x+1}+1\right|+\left|\sqrt{x+1}-3\right|=\left|2\sqrt{x+1}-2\right|\)
Áp dụng BĐT trị tuyệt đối:
\(\left|\sqrt{x+1}+1\right|+\left|\sqrt{x+1}-3\right|\ge\left|\sqrt{x+1}+1+\sqrt{x+1}-3\right|=\left|2\sqrt{x+1}-2\right|\)
Dấu "=" xảy ra khi và chỉ khi \(\left(\sqrt{x+1}+1\right)\left(\sqrt{x+1}-3\right)\ge0\)
\(\Leftrightarrow\sqrt{x+1}-3\ge0\)
\(\Leftrightarrow x+1\ge9\)
\(\Leftrightarrow x\ge8\)
1) \(\sqrt{5-2x}=6\left(đk:x\le\dfrac{5}{2}\right)\)
\(\Leftrightarrow5-2x=36\)
\(\Leftrightarrow2x=-31\Leftrightarrow x=-\dfrac{31}{2}\left(tm\right)\)
2) \(\sqrt{2-x}=\sqrt{x+1}\left(đk:2\ge x\ge-1\right)\)
\(\Leftrightarrow2-x=x+1\)
\(\Leftrightarrow2x=1\Leftrightarrow x=\dfrac{1}{2}\left(tm\right)\)
3) \(\Leftrightarrow\sqrt{\left(2x+1\right)^2}=6\)
\(\Leftrightarrow\left|2x+1\right|=6\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+1=6\\2x+1=-6\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5}{2}\\x=-\dfrac{7}{2}\end{matrix}\right.\)
4) \(\sqrt{x^2-10x+25}=x-2\left(đk:x\ge2\right)\)
\(\Leftrightarrow\sqrt{\left(x-5\right)^2}=x-2\)
\(\Leftrightarrow\left|x-5\right|=x-2\)
\(\Leftrightarrow\left[{}\begin{matrix}x-5=x-2\left(x\ge5\right)\\x-5=2-x\left(2\le x< 5\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}5=2\left(VLý\right)\\x=\dfrac{7}{2}\left(tm\right)\end{matrix}\right.\)
Không biết sao bạn cho thêm \(x\in Z\) vào cuối câu nhỉ? Giải pt nghiệm nguyên lai pt vô tỉ à :v
Bài làm :
\(pt\Leftrightarrow\sqrt{\left(x+1\right)\left(x+2\right)}+\sqrt{\left(x-1\right)\left(x+1\right)}+6=3\sqrt{x+1}+2\sqrt{x+2}+2\sqrt{x-1}\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x+1}=a\\\sqrt{x-1}=b\\\sqrt{x+2}=c\end{matrix}\right.\)
\(pt\Leftrightarrow ac+ab+6=3a+2b+2c\)
\(\Leftrightarrow ac+ab+6-3a-2b-2c=0\)
\(\Leftrightarrow c\left(a-2\right)+b\left(a-2\right)-3\left(a-2\right)=0\)
\(\Leftrightarrow\left(a-2\right)\left(b+c-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}a=2\\b+c=3\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x+1}=2\\\sqrt{x-1}+\sqrt{x+2}=3\end{matrix}\right.\)
+) TH1: \(\sqrt{x+1}=2\)
\(\Leftrightarrow x+1=4\)
\(\Leftrightarrow x=3\) ( thỏa )
+) TH2: \(\sqrt{x-1}+\sqrt{x+2}=3\)
\(\Leftrightarrow x-1+x+2+2\sqrt{\left(x-1\right)\left(x+2\right)}=9\)
\(\Leftrightarrow2\sqrt{\left(x-1\right)\left(x+2\right)}=8-2x\)
\(\Leftrightarrow\sqrt{\left(x-1\right)\left(x+2\right)}=4-x\)
\(\Leftrightarrow\left(x-1\right)\left(x+2\right)=\left(4-x\right)^2\)
\(\Leftrightarrow x^2+x-2=x^2-8x+16\)
\(\Leftrightarrow9x=18\)
\(\Leftrightarrow x=2\) ( thỏa )
Vậy \(x\in\left\{2;3\right\}\).
Điều kiện:`x>=2`
Ta có:
`sqrt{x+6}-sqrt{x-2}=(x+6-x+2)/(sqrt{x+6}+sqrt{x-2})`
`=8/(\sqrt{x+6}+sqrt{x-2})`
`pt<=>8/(sqrt{x+6}+sqrt{x-2})(1+sqrt{(x-2)(x+6)})=8`
`<=>(1+sqrt{(x-2)(x+6)})/(sqrt{x+6}+sqrt{x-2})=1`
`<=>1+sqrt{(x-2)(x+6)}=sqrt{x+6}+sqrt{x-2}`
`<=>sqrt{(x-2)(x+6)}-sqrt{x+6}=sqrt{x-2}-1`
`<=>sqrt{x+6}(sqrt{x-2}-1)=sqrt{x-2}-1`
`<=>(sqrt{x-2}-1)(sqrt{x+6}-1)=0`
Vì `x>=2=>x+6>=8=>sqrt{x+6}>=2sqrt2`
`=>sqrt{x+6}-1>=2sqrt2-1>0`
`<=>sqrt{x-2}=1`
`<=>x=3(tm)`
Vậy `S={3}`
a, \(\sqrt{2x^2-3}=\sqrt{4x-3}\) (x \(\ge\) \(\sqrt{\dfrac{3}{2}}\))
Vì hai vế ko âm, bp 2 vế ta được:
2x2 - 3 = 4x - 3
\(\Leftrightarrow\) 2x2 = 4x
\(\Leftrightarrow\) x2 = 2x
\(\Leftrightarrow\) x2 - 2x = 0
\(\Leftrightarrow\) x(x - 2) = 0
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x=0\\x-2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\left(KTM\right)\\x=2\left(TM\right)\end{matrix}\right.\)
Vậy S = {2}
b, \(\sqrt{2x-1}=\sqrt{x-1}\) (x \(\ge\) 1)
Vì hai vế ko âm, bp 2 vế ta được:
2x - 1 = x - 1
\(\Leftrightarrow\) x = 0 (KTM)
Vậy x = \(\varnothing\)
c, \(\sqrt{x^2-x-6}=\sqrt{x-3}\) (x \(\ge\) 3)
Vì hai vế ko âm, bp 2 vế ta được:
x2 - x - 6 = x - 3
\(\Leftrightarrow\) x2 - 2x - 3 = 0
\(\Leftrightarrow\) x2 - 3x + x - 3 = 0
\(\Leftrightarrow\) x(x - 3) + (x - 3) = 0
\(\Leftrightarrow\) (x - 3)(x + 1) = 0
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x-3=0\\x+1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\left(TM\right)\\x=-1\left(KTM\right)\end{matrix}\right.\)
Vậy S = {3}
d, \(\sqrt{x^2-x}=\sqrt{3x-5}\) (x \(\ge\) \(\dfrac{5}{3}\))
Vì hai vế ko âm, bp 2 vế ta được:
x2 - x = 3x - 5
\(\Leftrightarrow\) x2 - 4x + 5 = 0
\(\Leftrightarrow\) x2 - 4x + 4 + 1 = 0
\(\Leftrightarrow\) (x - 2)2 + 1 = 0
Vì (x - 2)2 \(\ge\) 0 với mọi x \(\ge\) \(\dfrac{5}{3}\) \(\Rightarrow\) (x - 2)2 + 1 > 0 với mọi x \(\ge\) \(\dfrac{5}{3}\)
\(\Rightarrow\) Pt vô nghiệm
Vậy S = \(\varnothing\)
Chúc bn học tốt!