\(\left(x+1\right)^{2010}+\left(x+2\right)^{2010}=2^{-2009}\)
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12 tháng 3 2017
đặt 2009-x=a,x-2010=b
suy ra a^2+ab+b^2/a^2-ab+b^2=19/49
suy ra 49(a^2+ab+b^2)=19(a^2-ab+b^2)
49a^2+49ab+49b^2=19a^2-19ab+19b^2
30a^2+68ab+30b^2=0
30a^2+50ab+18ab+30b^2=0
10a(3a+5b)+6b(3a+5b)=0
(3a+5b)(10a+6b)=0
suy ra 3a+5b=0 hoặc 10a+6b=0
thế vào lại rồi tìm x
8 tháng 1 2018
Đặt \(\left\{{}\begin{matrix}x-2010=a\\2009-x=b\end{matrix}\right.\)
Theo đề bài ta có:
\(\dfrac{\left(2009-x\right)^2+\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}{\left(2009-x\right)^2-\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}=\dfrac{19}{49}\)
\(\Leftrightarrow\dfrac{b^2+ab+a^2}{b^2-ab+a^2}=\dfrac{19}{49}\)
\(\Leftrightarrow19\left(b^2-ab+a^2\right)=49\left(b^2+ab+a^2\right)\)
\(\Leftrightarrow19b^2-19ab+19a^2-49b^2-49ab-49a^2=0\)
\(\Leftrightarrow-30a^2-68ab-30b^2=0\)
\(\Leftrightarrow-2\left(15a^2+34ab+15b^2\right)=0\)
\(\Leftrightarrow15a^2+34ab+15b^2=0\)
\(\Leftrightarrow15a^2+25ab+9ab+15b^2=0\)
\(\Leftrightarrow5a\left(3a+5b\right)+3b\left(3a+5b\right)=0\)
\(\Leftrightarrow\left(3a+5b\right)\left(5a+3b\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}3a+5b=0\\5a+3b=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}3\left(x-2010\right)+5\left(2009-x\right)=0\\5\left(x-2010\right)+3\left(2009-x\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}3x-6030+10045-5x=0\\5x-10050+6027-3x=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}-2x+4015=0\\2x-4023=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}-2x=-4015\\2x=4023\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{-4015}{-2}=2007,5\\x=\dfrac{4023}{2}=2011,5\end{matrix}\right.\)
Vậy \(\left[{}\begin{matrix}x=2007,5\\x=2011,5\end{matrix}\right.\)
MS
28 tháng 12 2017
Đặt a=(2009-x)2
b=(x-2010)2
Theo đề bài ta có
\(\dfrac{\text{a^2+ab+b^2}}{a^2-ab+b^2}=\dfrac{19}{49}\)
\(\text{49(a^2+ab+b^2)}=19\left(a^2-ab+b^2\right)\)
\(\text{30a^2+68ab+30b^2=0}\)
\(\text{15a^2+34ab+15b^2=0}\)
\(\text{15a^2+9ab+25ab+15b^2=0}\)
\(\text{3a(5a+3b)+5(3b+5a)=0}\)
\(\text{(5a+3b)(3a+5b)=0}\)
\(\left[{}\begin{matrix}3a+5b=0\\3b+5a=0\end{matrix}\right.\)
\(\left[{}\begin{matrix}3\left(2009-x\right)=5\left(x-2010\right)\\5\left(2009-x\right)=3\left(x-2010\right)\end{matrix}\right.\)
\(-8x=-6030-10045\) hay \(8x=-10050-6027\)
\(x\simeq2009\),375 hay \(x\simeq2009,625\)
21 tháng 12 2022
Đặt x-2009=a\(\Leftrightarrow\dfrac{\left(x-2009\right)^2-\left(x-2009\right)\left(x-2010\right)+\left(x-2010\right)^2}{\left(x-2009\right)^2+\left(x-2009\right)\left(x-2010\right)+\left(x-2010\right)^2}=\dfrac{19}{49}\)
\(\Leftrightarrow\dfrac{a^2-a\left(a-1\right)+\left(a-1\right)^2}{a^2+a\left(a-1\right)+\left(a-1\right)^2}=\dfrac{19}{49}\)
\(\Leftrightarrow\dfrac{a^2-a^2+a+a^2-2a+1}{a^2+a^2-a+a^2-2a+1}=\dfrac{19}{49}\)
=>\(\dfrac{a^2-a+1}{3a^2-3a+1}=\dfrac{19}{49}\)
=>49a^2-49a+49-57a^2+57a-19=0
=>-8a^2+8a+30=0
=>a=5/2 hoặc a=-3/2
=>x-2009=5/2 hoặc x-2009=-3/2
=>x=4023/2 hoặc x=4015/2
DP
4 tháng 4 2018
ĐKXĐ: \(x\ne2009\) ; \(x\ne2010\)
Đặt : a = \(2009-x\)
b = \(x-2010\)
⇒ a + b = -1 ⇒ a = - ( 1 + b )
⇒ Phương trình đã cho có dạng :
\(\dfrac{a^2+ab+b^2}{a^2-ab+b^2}=\dfrac{19}{49}\)
⇔ \(\dfrac{\left(1+b\right)^2-b\left(1+b\right)+b^2}{\left(1+b\right)^2+b\left(1+b\right)+b^2}\) \(=\dfrac{19}{49}\)
⇔ \(\dfrac{b^2+b+1}{3b^2+3b+1}\) \(=\dfrac{19}{49}\)
⇔ \(49b^2+49b+49=57b^2+57b+19\)
⇔ \(8b^2+8b-30=0\)
⇔ \(4b^2+4b-15=0\)
⇔ \(4b^2-6b+10b-15=0\)
⇔ \(2b\left(2b-3\right)+5\left(2b-3\right)=0\)
⇔ \(\left(2b-3\right)\left(2b+5\right)=0\)
⇔ \(\left[{}\begin{matrix}2b+5=0\\2b-3=0\end{matrix}\right.\) ⇔ \(\left[{}\begin{matrix}b=\dfrac{-5}{2}\\b=\dfrac{3}{2}\end{matrix}\right.\) ⇔ \(\left[{}\begin{matrix}x=\dfrac{-5}{2}+2010=2007,5\\x=\dfrac{3}{2}+2010=2011,5\end{matrix}\right.\)
Vậy ......
TT
21 tháng 2 2018
Đặt \(x-2009=y\) khi đó phương trình trở thành:
\(\dfrac{y^2-y\left(y-1\right)+\left(y-1\right)^2}{y^2+y\left(y-1\right)+\left(y-1\right)^2}=\dfrac{19}{49}\)
\(\Leftrightarrow4y^2-4y-15=0\)
\(\Leftrightarrow\left(2y-5\right)\left(2y+3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}y=2,5\\y=-1,5\end{matrix}\right.\)
Đổi lại:\(y=x-2009\) ,ta được:
\(\left[{}\begin{matrix}x=2009+2,5=2011,5\\x=2009-1,5=2007,5\end{matrix}\right.\)
Vậy...
