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4 tháng 4 2022
d. Áp dụng BĐT Caushy Schwartz ta có:
\(x+y+\dfrac{1}{x}+\dfrac{1}{y}\le x+y+\dfrac{\left(1+1\right)^2}{x+y}=x+y+\dfrac{4}{x+y}\le1+\dfrac{4}{1}=5\)
-Dấu bằng xảy ra \(\Leftrightarrow x=y=\dfrac{1}{2}\)
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
18 tháng 7 2023
a, (\(x-2\))2 - (2\(x\) + 3)2 = 0
(\(x\) - 2 - 2\(x\) - 3)(\(x\) - 2 + 2\(x\) + 3) = 0
(-\(x\) - 5)(3\(x\) +1) = 0
\(\left[{}\begin{matrix}-x-5=0\\3x+1=0\end{matrix}\right.\)
\(\left[{}\begin{matrix}x=-5\\3x=-1\end{matrix}\right.\)
\(\left[{}\begin{matrix}x=-5\\x=-\dfrac{1}{3}\end{matrix}\right.\)
Vậy \(x\in\) { -5;- \(\dfrac{1}{3}\)}
b, 9.(2\(x\) + 1)2 - 4.(\(x\) + 1)2 = 0
{3.(2\(x\) + 1) - 2.(\(x\) +1)}{ 3.(2\(x\) +1) + 2.(\(x\) +1)} = 0
(6\(x\) + 3 - 2\(x\) - 2)(6\(x\) + 3 + 2\(x\) + 2) = 0
(4\(x\) + 1)(8\(x\) + 5) =0
\(\left[{}\begin{matrix}4x+1=0\\8x+5=0\end{matrix}\right.\)
\(\left[{}\begin{matrix}x=-\dfrac{1}{4}\\x=-\dfrac{5}{8}\end{matrix}\right.\)
S = { - \(\dfrac{5}{8}\); \(\dfrac{-1}{4}\)}
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
18 tháng 7 2023
d, \(x^2\)(\(x\) + 1) - \(x\) (\(x+1\)) + \(x\)(\(x\) -1) = 0
\(x\left(x+1\right)\).(\(x\) - 1) + \(x\)(\(x\) -1) = 0
\(x\)(\(x\) -1)(\(x\) + 1 + 1) = 0
\(x\left(x-1\right)\left(x+2\right)\) = 0
\(\left[{}\begin{matrix}x=0\\x-1=0\\x+2=0\end{matrix}\right.\)
\(\left[{}\begin{matrix}x=0\\x=1\\x=-2\end{matrix}\right.\)
S = { -2; 0; 1}
MD
27 tháng 8 2021
a) 4x(x+1)=8(x+1)
<=>4x(x+1)-8(x+1)=0
<=>(4x-8)(x+1)=0
<=>\(\left[\begin{array}{} 4x-8=0\\ x+1=0 \end{array} \right.\)
<=>\(\left[\begin{array}{} x=2\\ x=-1 \end{array} \right.\)
Vậy...
b)x(x-1)-2(1-x)=0
<=>(x+2)(x-1)=0
<=>\(\left[\begin{array}{} x+2=0\\ x-1=0 \end{array} \right.\)
<=>\(\left[\begin{array}{} x=-2\\ x=1 \end{array} \right.\)
Vậy...
c)5x(x-2)-(2-x)=0
<=>(5x+1)(x-2)=0
<=>\(\left[\begin{array}{} 5x+1=0\\ x-2 \end{array} \right.\)
<=>\(\left[\begin{array}{} x=-1/5\\ x=2 \end{array} \right.\)
d)5x(x-200)-x+200=0
<=>(5x-1)(x-200)=0
<=>\(\left[\begin{array}{} 5x-1=0\\ x-200=0 \end{array} \right.\)
<=>\(\left[\begin{array}{} x=1/5\\ x=200 \end{array} \right.\)
e)\(x^3+4x=0 \)
\(\Leftrightarrow x(x^2+4)=0 \)
\(\Leftrightarrow \left[\begin{array}{} x=0\\ x^2+4=0 (loại vì x^2+4>=0 với mọi x) \end{array} \right.\)
Vậy x=0
f)\((x+1)=(x+1)^2\)
\(\Leftrightarrow (x+1)-(x+1)^2=0\)
\(\Leftrightarrow (x+1)(1-x-1)=0\)
\(\Leftrightarrow (x+1)(-x)=0\)
\(\Leftrightarrow \left[\begin{array}{} x=-1\\ x=0 \end{array} \right.\)
Vậy....
ND
Nguyễn Đức Trí
VIP
18 tháng 7 2023
a)\(\left(x-2\right)^2-\left(2x+3\right)^2=0\Rightarrow\left(x-2+2x+3\right)\left(x-2-2x-3\right)=0\)
\(\Rightarrow\left(3x+1\right)\left(-x-5\right)=0\Rightarrow\left[{}\begin{matrix}3x+1=0\\-x-5=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=-\dfrac{1}{3}\\x=-5\end{matrix}\right.\)
b)\(9\left(2x+1\right)^2-4\left(x+1\right)^2=0\Rightarrow\left[3\left(2x+1\right)+2\left(x+1\right)\right]\left[3\left(2x+1\right)-2\left(x+1\right)\right]=0\)
\(\Rightarrow\left[8x+5\right]\left[4x+1\right]=0\Rightarrow\left[{}\begin{matrix}8x+5=0\\4x-1=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=-\dfrac{5}{8}\\x=\dfrac{1}{4}\end{matrix}\right.\)
c)\(x^3-6x^2+9x=0\Rightarrow x\left(x^2-6x+9\right)=0\Rightarrow x\left(x-3\right)^2=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x-3=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)
d) \(x^2\left(x+1\right)-x\left(x+1\right)+x\left(x-1\right)=0\)
\(\Rightarrow x\left(x+1\right)\left(x^2-1\right)+x\left(x-1\right)=0\)
\(\Rightarrow x\left(x+1\right)\left(x-1\right)\left(x+1\right)+x\left(x-1\right)=0\)
\(\Rightarrow x\left(x-1\right)\left[\left(x+1\right)\left(x+1\right)+1\right]=0\)
\(\Rightarrow x\left(x-1\right)\left[\left(x+1\right)^2+1\right]=0\)
Do \(\left(x+1\right)^2+1>0\)
\(\Rightarrow x\left(x-1\right)=0\Rightarrow\left[{}\begin{matrix}x=0\\x-1=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)
H
3
21 tháng 8 2021
b) \(x^3-x^2-x+1=0\Leftrightarrow\left(x-1\right)^2\left(x+1\right)=0\)
\(\Leftrightarrow x-1=0\) hoặc \(x+1=0\)
\(\Leftrightarrow x=1\) hoặc \(x=-1\)
c) \(x^2-6x+8=0\Leftrightarrow\left(x-4\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-4=0\\x-2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=4\\x=2\end{matrix}\right.\)
21 tháng 8 2021
a) \(x^3+x^2+x+1=0\Leftrightarrow\left(x+1\right)\left(x^2+1\right)=0\Leftrightarrow x+1=0\Leftrightarrow x=-1\)
(do \(x^2+1\ge1>0\))
31 tháng 7 2020
a) (x+2)(x+1-x+1)=0
\(\Leftrightarrow\) (x+2)\(\times\) 2 = 0
\(\)\(\Leftrightarrow\)x+2 =0\(\Leftrightarrow\) x =-2
b) \(x^3-2x^2+x^2+x-2\)
\(\Leftrightarrow x^3-x^2+x-2=0\)
\(\left(x+1\right)\left(x+1\right)^2=0\)
\(\Leftrightarrow\left(x+1\right)^3=0\)
\(\Leftrightarrow x+1=0\)
\(\Leftrightarrow x=-1\)
Vậy \(x=-1\)
\(\left(x+1\right)\left(x+1\right)^2=0\)
\(\left(x+1\right)\left(x+1+1\right)=0\)
\(\left(x+1\right)\left(x+2\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x+1=0\\x+2=0\end{cases}\Rightarrow\orbr{\begin{cases}x=-1\\x=-2\end{cases}}}\)