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Gợi ý :
Bài 1 : Cộng thêm 1 vào 3 phân thức đầu, trừ cho 3 ở phân thức thứ 4, có nhân tử chung là (x+2020)
Bài 2 : Trừ mỗi phân thức cho 1, chuyển vế và có nhân tử chung là (x-2021)
Bài 3 : Phân thức thứ nhất trừ đi 1, phân thức hai trù đi 2, phân thức ba trừ đi 3, phân thức bốn trừ cho 4, phân thức 5 trừ cho 5. Có nhân tử chung là (x-100)
bài 3
\(\frac{x-90}{10}+\frac{x-76}{12}+\frac{x-58}{14}+\frac{x-36}{16}+\frac{x-15}{17}=15.\)
=>\(\frac{x-90}{10}-1+\frac{x-76}{12}-2+\frac{x-58}{14}-3+\frac{x-36}{16}-4+\frac{x-15}{17}-5=0\)
=>\(\frac{x-100}{10}+\frac{x-100}{12}+\frac{x-100}{14}+\frac{x-100}{16}+\frac{x-100}{17}=0\)
=>\(\left(x-100\right).\left(\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+\frac{1}{16}+\frac{1}{17}\right)=0\)
=>(x-100)=0 do \(\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+\frac{1}{16}+\frac{1}{17}\ne0\)
=> x=100
\(\frac{x+1}{2018}+\frac{x+2}{2019}=\frac{x+3}{2020}+\frac{x+4}{2021}\)
\(\Leftrightarrow\left(\frac{x+1}{2018}-1\right)+\left(\frac{x+2}{2019}-1\right)=\left(\frac{x+3}{2020}-1\right)+\left(\frac{x+4}{2021}-1\right)\)
\(\Leftrightarrow\frac{x-2017}{2018}+\frac{x-2017}{2019}=\frac{x-2017}{2020}+\frac{x-2017}{2021}\)
\(\Leftrightarrow\left(x-2017\right)\left(\frac{1}{2018}+\frac{1}{2019}-\frac{1}{2020}-\frac{1}{2021}\right)=0\)
\(\Leftrightarrow x-2017=0\)\(\left(\frac{1}{2018}+\frac{1}{2019}-\frac{1}{2020}-\frac{1}{2021}\ne0\right)\)
\(\Leftrightarrow x=2017\)
Vậy \(S=\left\{2017\right\}\)
Ta có: \(\frac{x-5}{2016}+\frac{x-4}{2017}=\frac{x-3}{2018}+\frac{x-2}{2019}\)
\(\Leftrightarrow\frac{x-5}{2016}-1+\frac{x-4}{2017}-1=\frac{x-3}{2018}-1+\frac{x-2}{2019}-1\)
\(\Leftrightarrow\frac{x-2021}{2016}+\frac{x-2021}{2017}=\frac{x-2021}{2018}+\frac{x-2021}{2019}\)
\(\Leftrightarrow\frac{x-2021}{2016}+\frac{x-2021}{2017}-\frac{x-2021}{2018}-\frac{x-2021}{2019}=0\)
\(\Leftrightarrow\left(x-2021\right)\left(\frac{1}{2016}+\frac{1}{2017}-\frac{1}{2018}-\frac{1}{2019}\right)=0\)
mà \(\frac{1}{2016}+\frac{1}{2017}-\frac{1}{2018}-\frac{1}{2019}\ne0\)
nên x-2021=0
hay x=2021
Vậy: x=2021
Giải các pt sau:
a) (x+4)(2x-3)=0
TH1: x+4=0 => x=-4
TH2 : 2x-3=0 => 2x=3 =>x=3/2
Ta có \(\frac{2015}{2016}.x+\frac{2016}{2017}.x+\frac{2017}{2018}.x=\frac{2018}{2019}.x\)
<=>\(\frac{2015}{2016}.x+\frac{2016}{2017}.x+\frac{2017}{2018}x-\frac{2018}{2019}x=0\)
<=>x\(\left(\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}-\frac{2018}{2019}\right)=0\)
Vì \(\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}-\frac{2018}{2019}\) không thể bằng 0
Vậy x=0
Ta có 1 nghiệm thỏa mãn S=\(\left\{0\right\}\)
B. \(\frac{x+4}{2015}+1+\frac{x+3}{2016}+1=\frac{x+2}{2017}+1+\frac{x+1}{2018}+1\)
<=> \(\frac{x+2019}{2015}+\frac{x+2019}{2016}=\frac{x+2019}{2017}+\frac{x+2019}{2018}\)
<=>(x+2019).(\(\frac{1}{2015}+\frac{1}{2016}-\frac{1}{2017}-\frac{1}{2018}>0\)
Vì (\(\frac{1}{2015}+\frac{1}{2016}-\frac{1}{2017}-\frac{1}{2018}>0\)
=> x+2019>0
=>x>-2019
d, \(\frac{x+1}{9}+\frac{x+2}{8}=\frac{x+3}{7}+\frac{x+4}{6}\)
\(\Leftrightarrow\frac{x+1}{9}+1+\frac{x+2}{8}+1=\frac{x+3}{7}+1+\frac{x+4}{6}+1\)
\(\Leftrightarrow\frac{x+10}{9}+\frac{x+10}{8}-\frac{x+10}{7}-\frac{x+10}{6}=0\)
\(\Leftrightarrow\left(x+10\right)\left(\frac{1}{9}+\frac{1}{8}-\frac{1}{7}-\frac{1}{6}\right)=0\)
\(\Leftrightarrow x+10=0\) (Vì \(\frac{1}{9}+\frac{1}{8}-\frac{1}{7}-\frac{1}{6}\) ≠ 0)
\(\Leftrightarrow x=-10\)
Vậy x = -10 là nghiệm của phương trình.
Bài 3 :
Ta có : \(A=x^2+x+2012\)
=> \(A=x^2+x+\left(\frac{1}{2}\right)^2+\frac{8047}{4}\)
=> \(A=\left(x+\frac{1}{2}\right)^2+\frac{8047}{4}\)
- Ta thấy : \(\left(x+\frac{1}{2}\right)^2\ge0\forall x\)
=> \(\left(x+\frac{1}{2}\right)^2+\frac{8047}{4}\ge\frac{8047}{4}\forall x\)
- Dấu "=" xảy ra <=> \(x+\frac{1}{2}=0\)
<=> \(x=-\frac{1}{2}\)
Vậy MinA = \(\frac{8047}{4}\) <=> x = \(-\frac{1}{2}\) .
Bài 1 :
a, Ta có : \(\left(3x-2\right)\left(4+5x\right)=0\)
=> \(\left[{}\begin{matrix}3x-2=0\\4+5x=0\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}3x=2\\5x=-4\end{matrix}\right.\)
=> \(\left[{}\begin{matrix}x=\frac{2}{3}\\x=-\frac{4}{5}\end{matrix}\right.\)
Vậy phương trình có nghiệm là x = \(\frac{2}{3}\), x = \(-\frac{4}{5}\) .
b,- ĐKXĐ : \(\left\{{}\begin{matrix}x-1\ne0\\x+1\ne0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x\ne1\\x\ne-1\end{matrix}\right.\)
=> \(x\ne\pm1\)
Ta có : \(\frac{x+1}{x-1}-\frac{4}{x+1}=\frac{3-x^2}{1-x^2}\)
=> \(\frac{\left(x+1\right)^2}{x^2-1}-\frac{4\left(x-1\right)}{x^2-1}=\frac{x^2-3}{x^2-1}\)
=> \(\left(x+1\right)^2-4\left(x-1\right)=x^2-3\)
=> \(x^2+2x+1-4x+4=x^2-3\)
=> \(-2x=-3-5\)
=> \(x=4\left(TM\right)\)
Vậy phương trình có nghiệm là x = 4 .
c, Ta có : \(\frac{10x+3}{2009}+\frac{10x-1}{2013}=\frac{10x+1}{2011}-\frac{2-10x}{2014}\)
=> \(\frac{10x+3}{2009}+\frac{10x-1}{2013}=\frac{10x+1}{2011}+\frac{10x-2}{2014}\)
=> \(\frac{10x+3}{2009}+1+\frac{10x-1}{2013}+1=\frac{10x+1}{2011}+1+\frac{10x-2}{2014}+1\)
=> \(\frac{10x+3}{2009}+\frac{2009}{2009}+\frac{10x-1}{2013}+\frac{2013}{2013}=\frac{10x+1}{2011}+\frac{2011}{2011}+\frac{10x-2}{2014}+\frac{2014}{2014}\)
=> \(\frac{10x+2012}{2009}+\frac{10x+2012}{2013}=\frac{10x+2012}{2011}+\frac{10x+2012}{2014}\)
=> \(\frac{10x+2012}{2009}+\frac{10x+2012}{2013}-\frac{10x+2012}{2011}-\frac{10x+2012}{2014}=0\)
=> \(\left(10x+2012\right)\left(\frac{1}{2009}+\frac{1}{2013}-\frac{1}{2011}-\frac{1}{2014}\right)=0\)
=> \(10x+2012=0\)
=> \(x=-\frac{2012}{10}\)
Vậy phương trình có nghiệm là x = \(-\frac{2012}{10}\) .
Bài 3:
Giải:
Ta có : A = x2 + x + 2012
= x2 + 2.\(\frac{1}{2}\).x + \(\frac{1}{4}\) + \(\frac{8047}{4}\)
= (x + \(\frac{1}{2}\))2 + \(\frac{8047}{4}\) ≥ \(\frac{8047}{4}\)
⇒ Amin = \(\frac{8047}{4}\) ⇔ (x + \(\frac{1}{2}\))2 = 0 ⇔ x = \(-\frac{1}{2}\)
Vậy Amin = \(\frac{8047}{4}\) tại x = \(-\frac{1}{2}\)
Chúc bạn học tốt@@
Ta có\(\frac{x+2052}{30}+\frac{x+2014}{4}+\frac{x+47}{1975}=\frac{x+4}{2018}+\frac{x+3}{2019}\)
<=> \(\left(\frac{x+2052}{30}-1\right)+\left(\frac{x+2014}{4}+2\right)+\left(\frac{x+47}{1975}+1\right)=\left(\frac{x+4}{2018}+1\right)+\left(\frac{x+3}{2019}+1\right)\)
<=> \(\frac{x+2022}{30}+\frac{x+2022}{4}+\frac{x+2022}{1975}=\frac{x+2022}{2018}+\frac{x+2022}{2019}\)
<=> \(\frac{x+2022}{30}+\frac{x+2022}{4}+\frac{x+2022}{1975}-\frac{x+2022}{2018}-\frac{x+2022}{2019}=0\)
<=> \(\left(x+2022\right)\left(\frac{1}{30}+\frac{1}{4}+\frac{1}{1975}-\frac{1}{2018}-\frac{1}{2019}\right)=0\)
<=> x + 2022 = 0 (Vì \(\frac{1}{30}+\frac{1}{4}+\frac{1}{1975}-\frac{1}{2018}-\frac{1}{2019}\ne0\))
<=> x = -2022
Vậy x = -2022 là nghiệm phương trình
\(\frac{x+2052}{30}+\frac{x+2014}{4}+\frac{x+47}{1975}=\frac{x+4}{2018}+\frac{x+3}{2019}\)
\(\Leftrightarrow\left(\frac{x+2052}{30}-1\right)+\left(\frac{x+2014}{4}+2\right)+\left(\frac{x+47}{1975}+1\right)=\left(\frac{x+4}{2018}+1\right)+\left(\frac{x+3}{2019}+1\right)\)
\(\Leftrightarrow\frac{x+2052-30}{30}+\frac{x+2014+8}{4}+\frac{x+47+1975}{1975}=\frac{x+4+2018}{2018}+\frac{x+3+2019}{2019}\)
\(\Leftrightarrow\frac{x+2022}{30}+\frac{x+2022}{4}+\frac{x+2022}{1975}=\frac{x+2022}{2018}+\frac{x+2022}{2019}\)
\(\Leftrightarrow\left(x+2022\right)\left(\frac{1}{30}+\frac{1}{4}+\frac{1}{1975}-\frac{1}{2018}-\frac{1}{2019}\right)=0\)
\(\Leftrightarrow x+2022=0\)
\(\Leftrightarrow x=-2022\)