giải pt x2 -\(\sqrt{x-5}\)= 5
K
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Y
24 tháng 6 2023
\(\left(2-\sqrt{5}\right)x^2+\left(6-\sqrt{5}\right)x-8+2\sqrt{5}=0\)
\(\Leftrightarrow\left(2-\sqrt{5}\right)x^2-\left(2-\sqrt{5}\right)x+\left(8-2\sqrt{5}\right)x-(8-2\sqrt{5})=0\)
\(\Leftrightarrow\left(2-\sqrt{5}\right)x\left(x-1\right)+\left(8-2\sqrt{5}\right)\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left[\left(2-\sqrt{5}\right)x+\left(8-2\sqrt{5}\right)\right]=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-1=0\\\left(2-\sqrt{5}\right)x=-8+2\sqrt{5}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=\dfrac{-8+2\sqrt{5}}{2-\sqrt{5}}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=6+4\sqrt{5}\end{matrix}\right.\)
Vậy \(S=\left\{1;6+4\sqrt{5}\right\}\)
NT
0
K
2
NV
Nguyễn Việt Lâm
Giáo viên
12 tháng 12 2020
ĐKXĐ: \(x\ge-5\)
\(\Leftrightarrow\left(x+7\right)^2-2\left(x+7\right)\sqrt{x+5}+x+5-16=0\)
\(\Leftrightarrow\left(x+7-\sqrt{x+5}\right)^2-16=0\)
\(\Leftrightarrow\left(x+7-\sqrt{x+5}-4\right)\left(x+7-\sqrt{x+5}+4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x+5}=x+3\left(x\ge-3\right)\\\sqrt{x+5}=x+11\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x+5=x^2+6x+9\\x+5=x^2+22x+121\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+5x+4=0\\x^2+21x+116=0\left(vn\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=-4< -3\left(l\right)\end{matrix}\right.\)
NV
Nguyễn Việt Lâm
Giáo viên
5 tháng 10 2021
ĐKXĐ: \(x\ge-1\)
\(5\sqrt{\left(x+1\right)\left(x^2-x+1\right)}=2\left(x^2+2\right)\)
Đặt \(\left\{{}\begin{matrix}\sqrt{x+1}=a\ge0\\\sqrt{x^2-x+1}=b>0\end{matrix}\right.\)
\(\Rightarrow a^2+b^2=x^2+2\)
Phương trình trở thành:
\(5ab=2\left(a^2+b^2\right)\)
\(\Leftrightarrow2a^2-5ab+2b^2=0\)
\(\Leftrightarrow\left(2a-b\right)\left(a-2b\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}2a=b\\a=2b\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}2\sqrt{x+1}=\sqrt{x^2-x+1}\\\sqrt{x+1}=2\sqrt{x^2-x+1}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}4\left(x+1\right)=x^2-x+1\\x+1=4\left(x^2-x+1\right)\end{matrix}\right.\)
\(\Leftrightarrow...\)
TS
1
13 tháng 5 2021
Ta có: \(\Delta=\left(2m-1\right)^2-4\cdot1\cdot\left(m^2-2\right)\)
\(=4m^2-4m+1-4m^2+8\)
\(=-4m+9\)
Để phương trình có hai nghiệm phân biệt thì \(\Delta>0\)
\(\Leftrightarrow-4m+9>0\)
\(\Leftrightarrow-4m>-9\)
hay \(m< \dfrac{9}{4}\)
Áp dụng hệ thức Vi-et, ta được:
\(\left\{{}\begin{matrix}x_1+x_2=2m-1\\x_1\cdot x_2=m^2-2\end{matrix}\right.\)
Ta có: \(\left|x_1-x_2\right|=\sqrt{5}\)
\(\Leftrightarrow\sqrt{\left(x_1-x_2\right)^2}=\sqrt{5}\)
\(\Leftrightarrow\sqrt{\left(x_1+x_2\right)^2-4x_1x_2}=\sqrt{5}\)
\(\Leftrightarrow\left(2m-1\right)^2-4\cdot\left(m^2-2\right)=5\)
\(\Leftrightarrow4m^2-4m+1-4m^2+8=5\)
\(\Leftrightarrow-4m=-4\)
hay m=1(thỏa ĐK)
Vậy: m=1
13 tháng 5 2021
PT có 2 nghiệm phân biệt
`<=>Delta>0`
`<=>(2m-1)^2-4(m^2-2)>0`
`<=>4m^2-4m+1-4m^2+8>0`
`<=>-4m+9>0`
`<=>m<9/4`
Áp dụng vi-ét:`x_1+x_2=2m-1,x_1.x_2=m^2-2`
`|x_1-x_2|=\sqrt5`
`<=>(x_1-x_2)^2=5`
`<=>(x_1+x_2)^2-4(x_1.x_2)=5`
`<=>4m^2-4m+1-4m^2+8=5`
`<=>-4m+8=5`
`<=>4m=3`
`<=>m=3/4(tm)`
Vậy `m=3/4=>|x_1-x_2|=\sqrt5`
VT
1
3 tháng 4 2022
<=>\(\left\{{}\begin{matrix}\sqrt{5}x-2y=7\\\sqrt{5}x-5y=10\end{matrix}\right.\)
<=>\(\left\{{}\begin{matrix}-3y=3\\\sqrt{5}x-2y=7\end{matrix}\right.\)
<=>\(\left\{{}\begin{matrix}y=-1\\x=\sqrt{5}\end{matrix}\right.\)
KL: vậy hpt có ngiệm là \(\left\{{}\begin{matrix}x=\sqrt{5}\\y=-1\end{matrix}\right.\)
NK
1
6 tháng 7 2021
Ta có: \(\dfrac{2x}{x^2-x+1}-\dfrac{x}{x^2+x+1}=\dfrac{5}{3}\)
\(\Leftrightarrow\dfrac{2x\left(x^2+x+1\right)-x\left(x^2-x+1\right)}{\left(x^2-x+1\right)\left(x^2+x+1\right)}=\dfrac{5}{3}\)
\(\Leftrightarrow\dfrac{2x^3+2x^2+2x-x^3+x^2-x}{\left(x^2-x+1\right)\left(x^2+x+1\right)}=\dfrac{5}{3}\)
\(\Leftrightarrow\dfrac{x^3+3x^2+x}{\left(x^2+1\right)^2-x^2}=\dfrac{5}{3}\)
\(\Leftrightarrow3x^3+9x^2+3x=5\left(x^4+2x^2+1-x^2\right)\)
\(\Leftrightarrow3x^3+9x^2+3x=5x^4+5x^2+5\)
\(\Leftrightarrow5x^4+5x^2+5-3x^3-9x^2-3x=0\)
\(\Leftrightarrow5x^4-3x^3-4x^2-3x+5=0\)
\(\Leftrightarrow5x^4-5x^3+2x^3-2x^2-2x^2+2x-5x+5=0\)
\(\Leftrightarrow5x^3\left(x-1\right)+2x^2\left(x-1\right)-2x\left(x-1\right)-5\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(5x^3+2x^2-2x-5\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(5x^3-5x^2+7x^2-7x+5x-5\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left[5x^2\left(x-1\right)+7x\left(x-1\right)+5\left(x-1\right)\right]=0\)
\(\Leftrightarrow\left(x-1\right)^2\cdot\left(5x^2+7x+5\right)=0\)
mà \(5x^2+7x+5>0\forall x\)
nên x-1=0
hay x=1
NA
0
Điều kiện: \(\hept{\begin{cases}x-5\ge0\\x^2-5\ge0\end{cases}}\)
\(\Leftrightarrow x\ge5\)
\(\Leftrightarrow\left(x^2-5\right)^2=x-5\)
\(\Leftrightarrow x^4-10x^2-x+30=0\)
\(\Leftrightarrow2x^4-20x^2-2x+60=0\)
\(\Leftrightarrow\left(2x^4-50x^2+625\right)+\left(x^2-2x+1\right)+\left(29x^2-725\right)+159=0\)
\(\Leftrightarrow2\left(x^2-25\right)^2+\left(x-1\right)^2+29\left(x^2-25\right)+159=0\)
Với \(x\ge5\)thì \(VP>0\)
Vậy PT vô nghiệm