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ĐKXĐ : Tự tìm nha : )
Ta có : \(\frac{1}{x^2+3x+2}+\frac{1}{x^2+5x+6}+...+\frac{1}{x^2+15x+56}=\frac{1}{14}\)
=> \(\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+...+\frac{1}{\left(x+7\right)\left(x+8\right)}=\frac{1}{14}\)
=> \(\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+...+\frac{1}{x+7}-\frac{1}{x+8}=\frac{1}{14}\)
=> \(\frac{1}{x+1}-\frac{1}{x+8}=\frac{1}{14}\)
=> \(\frac{x+8}{\left(x+1\right)\left(x+8\right)}-\frac{x+1}{\left(x+8\right)\left(x+1\right)}=\frac{1}{14}\)
=> \(14\left(x+8-x-1\right)=\left(x+1\right)\left(x+8\right)\)
=> \(x^2+x+8x+8=98\)
=> \(x^2+9x-90=0\)
=> \(\left(x+15\right)\left(x-6\right)=0\)
=> \(\left[{}\begin{matrix}x+15=0\\x-6=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x=-15\\x=6\end{matrix}\right.\) ( TM )
Vậy phương trình trên có nghiệm là \(S=\left\{6,-15\right\}\)
Phân tích mẫu thức thành nhân tử ta có :
1/(x+1)(x+2)+1/(x+2)(x+3)+...+1/(x+7)(x+8)=1/14
1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+...+1/(x+7)-1/(x+8)=1/14
1/(x+1)-1/(x+8)=1/14
7/(x+1)(x+8)=1/14
Nhân chéo ta có x^2+9x+8=98
x^2+9x-90=0
(x+15)(x-6)=0
Suy ra x=-15 hoặc x=6
Lời giải:
PT \(\Leftrightarrow \frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)}+\frac{1}{(x+3)(x+4)}+....+\frac{1}{(x+7)(x+8)}=\frac{1}{14}\)
(ĐK: $x\neq -1;-2;...;-8$)
\(\Leftrightarrow \frac{(x+2)-(x+1)}{(x+1)(x+2)}+\frac{(x+3)-(x+2)}{(x+2)(x+3)}+....+\frac{(x+8)-(x+7)}{(x+7)(x+8)}=\frac{1}{14}\)
\(\Leftrightarrow \frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+....+\frac{1}{x+7}-\frac{1}{x+8}=\frac{1}{14}\)
\(\Leftrightarrow \frac{1}{x+1}-\frac{1}{x+8}=\frac{1}{14}\Leftrightarrow \frac{7}{x^2+9x+8}=\frac{1}{14}\)
\(\Rightarrow x^2+9x+8=98\Leftrightarrow x^2+9x-90=0\Rightarrow x=6\) hoặc $x=-15$ (đều thỏa mãn)
Vậy........
Ta có : \(\frac{1}{x^2+3x+2}+\frac{1}{x^2+5x+6}+...+\) \(\frac{1}{x^2+15x+56}=\frac{1}{14}\)
<=>\(\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}\)+...+ \(\frac{1}{\left(x+7\right)\left(x+8\right)}=\frac{1}{14}\)
<=> \(\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+...+\frac{1}{x+7}-\frac{1}{x+8}\)= \(\frac{1}{14}\)
<=> \(\frac{1}{x+1}-\frac{1}{x+8}=\frac{1}{14}\)
<=> \(\frac{x+8-x-1}{\left(x+1\right)\left(x+8\right)}=\frac{1}{14}\)
<=>\(\frac{7.14}{14\left(x+1\right)\left(x+8\right)}=\frac{\left(x+1\right)\left(x+8\right)}{14\left(x+1\right)\left(x+8\right)}\)
<=> \(x^2+9x+8=98\)<=> \(x^2+9x-90=0\)
<=> (x-6)(x+15) =0
<=> \(\orbr{\begin{cases}x=6\\x=-15\end{cases}}\)
Vậy phương trình có 2 nghiệm x \(\in\left(6,15\right)\)
==============
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- Đề ko rõ ràng , lần sau nhớ ghi yêu cầu ?
\(\frac{1}{x^2+3x+2}+\frac{1}{x^2+5x+6}+...+\frac{1}{x^2+15x+56}=\frac{1}{14}\)
\(\Leftrightarrow\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+...+\frac{1}{\left(x+7\right)\left(x+8\right)}=\frac{1}{14}\)
\(\Leftrightarrow\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+...+\frac{1}{x+7}-\frac{1}{x+8}=\frac{1}{14}\)
\(\Leftrightarrow\frac{1}{x+1}-\frac{1}{x+8}=\frac{1}{14}\)
Làm nốt
2/
\(T=8x^2-4x+\frac{1}{4x^2}+15\)
\(=\left(4x^2-4x+1\right)+\left(4x^2+\frac{1}{4x^2}-2\right)+16\)
\(=\left(2x-1\right)^2+\left(\frac{4x^2-1}{2x}\right)^2+16\ge16\)
\(\frac{2x-1}{3x^2+7x+2}+\frac{3}{9x^2+15x+4}-\frac{2x+7}{3x^2-5x-12}=\frac{5}{x+2}\)
\(\Leftrightarrow\frac{2x-1}{\left(3x+1\right)\left(x+2\right)}+\frac{3}{\left(3x+1\right)\left(3x+4\right)}-\frac{2x+7}{\left(4x+3\right)\left(x-3\right)}=\frac{5}{\left(x+2\right)}\)
\(\Leftrightarrow\frac{1}{x+2}-\frac{1}{3x+1}+\frac{1}{3x+1}-\frac{1}{3x+4}+\frac{1}{3x+4}-\frac{1}{x-3}=\frac{5}{x+2}\)
\(\Leftrightarrow\frac{1}{x+2}-\frac{1}{x-3}=\frac{5}{x+2}\)
\(\Leftrightarrow\frac{x-3-x-2}{\left(x+2\right)\left(x-3\right)}=\frac{5\left(x-3\right)}{\left(x+2\right)\left(x-3\right)}\)
\(\Leftrightarrow5x-3=-5\)
\(\Leftrightarrow x=-\frac{2}{5}\)
Chúc bạn học tốt !!!
d) x+1/2019 + x+3/2017 = x+5/2015 + x+7/2013
<=> x+1/2019 + x+3/2017 - x+5/2015 - x+7/2013 =0
<=> ( x+1/2019 + 1) + ( x+3/2017 + 1) - ( x+5/2015 + 1) - ( x+7/2013 +1) = 0
<=> ( x+1+2019/2019) +(x+3+2017/2017) - ( x+5+2015/2015) - ( x+7+2013/2013) =0
<=> x+2020/2019 + x+2020/2017 - x+2020/2015 - x+2020/2013 =0
<=> (x+2020)× ( 1/2019 + 1/2017 - 1/2015 - 1/2013) =0
Mà 1/2019 + 1/2017 - 1/2015 - 1/2013 khác 0
=> x+2020 =0
=> x = -2020
\(\left(x-1\right)=\left(x-1\right)\left(x-2\right)\)
\(\Leftrightarrow\left(x-1\right)-\left(x-1\right)\left(x-2\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-2-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(x-3\right)=0\)
HOẶC\(x-1=0\Leftrightarrow x=1\)(NHẬN)
HOẶC\(x-3=0\Leftrightarrow x=3\)(NHẬN)
VẬY: tập ngiệm của pt là S={1;3}
\(\Leftrightarrow\frac{1}{\left(x-1\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-3\right)}+\frac{1}{\left(x-3\right)\left(x-4\right)}+\frac{1}{\left(x-4\right)\left(x-5\right)}=\frac{1}{8}\)
\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x-2}+\frac{1}{x-2}-\frac{1}{x-3}+...+\frac{1}{x-4}-\frac{1}{x-5}=\frac{1}{8}\)
\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x-5}=\frac{1}{8}\)
\(\Leftrightarrow\frac{x-5-x+1}{\left(x-1\right)\left(x-5\right)}=\frac{1}{8}\)
\(\Leftrightarrow-4.8=x^2-6x+5\)
\(\Leftrightarrow x^2-6x+37=0\)
Để PT đc xác định : \(x^2+3x+2\ne0;x^2+5x+6\ne0;.....;x^2+15x+56\ne0\)
\(\Leftrightarrow\left(x+1\right)\left(x+2\right)\ne0;\left(x+2\right)\left(x+3\right)\ne0;....;\left(x+7\right)\left(x+8\right)\ne0\)
\(\Rightarrow x+1;x+2;x+3;....;x+8\ne0\)
\(\Rightarrow x\ne\left\{-8;-7;...;-3;-2;-1\right\}\)
TXĐ : \(x\ne\left\{-8;-7;...;-3;-2;-1\right\}\)
\(\frac{1}{x^2+3x+2}+\frac{1}{x^2+5x+6}+....+\frac{1}{x^2+15x+56}=\frac{1}{14}\)
\(\Leftrightarrow\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+....+\frac{1}{\left(x+7\right)\left(x+8\right)}=\frac{1}{14}\)
\(\Leftrightarrow\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+...+\frac{1}{x+7}-\frac{1}{x+8}=\frac{1}{14}\)
\(\Leftrightarrow\frac{1}{x+1}-\frac{1}{x+8}=\frac{1}{14}\)
\(\Leftrightarrow\frac{7}{x^2+9x+8}=\frac{1}{14}\)
\(\Leftrightarrow x^2+9x+8=98\)
\(\Leftrightarrow x^2+9x-90=0\)
\(\Leftrightarrow\left(x-6\right)\left(x+15\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x=6\\x=-15\end{cases}}\)(TMĐKXĐ)
Vậy \(x=6\) hoặc \(x=-15\)