\(\frac{1}{x^2+9x+20}+\frac{1}{x^2+11x+30}+\frac{1}{x^2+13x+42}=\frac{1}{18}\)

<=> \(\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

<=>\(\frac{\left(x+6\right)\left(x+7\right)+\left(x+4\right)\left(x+7\right)+\left(x+4\right)\left(x+5\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

Từ đó, bạn tính ra nhá! Hơi dài, ai có cách nào ngắn hơn thì nói với mình nha!

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ĐK : \(\left(x\ne-4;x\ne-5;x\ne-6;x\ne-7\right)\)

\(\Rightarrow\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{x+4}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{x+4}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Rightarrow\frac{3}{x^2+11x+28}=\frac{1}{18}\)

\(\Leftrightarrow x^2+11x+28=54\)

\(\Rightarrow x^2+11x-26=0\)

\(\Rightarrow\left(x-2\right)\left(x+13\right)=0\)

\(\Rightarrow\orbr{\begin{cases}x=2\\x=-13\end{cases}}\)

Vậy pt có tập nghiệm là \(S=\left\{2;-13\right\}\)

Đk:\(\left(x\ne-4;x\ne-5;x\ne-6;x\ne-7\right)\)

\(\Rightarrow\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{x+4}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{x+4}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Rightarrow\frac{3}{x^2+11x+28}=\frac{1}{18}\)

\(\Leftrightarrow x^2+11x+28=54\)

\(\Rightarrow x^2+11x-26=0\)

\(\Rightarrow\left(x-2\right)\left(x+13\right)=0\)

\(\Rightarrow\left[\begin{array}{nghiempt}x=2\\x=-13\end{array}\right.\)

Vậy pt có tập nghiệm là S={2,-13}

Đk:

Vậy pt có tập nghiệm là S={2,-13}

\(\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}\)

\(\frac{1}{x+4}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+7}=\frac{1}{18}\)

\(\frac{1}{x+4}-\frac{1}{x+7}=\frac{1}{18}\)

dit me may

\(\frac{1}{x^2+9x+20}+\frac{1}{x^2+11x+30}+\frac{1}{x^2+13x+42}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{x+4}-\frac{1}{x+5}+\frac{1}{\left(x+5\right)}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Rightarrow\frac{1}{x+4}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Rightarrow\frac{3}{\left(x+4\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Rightarrow x^2+11x+28=54\)

\(\Rightarrow x^2+11x+\frac{121}{4}-\frac{9}{4}=54\)

\(\Rightarrow\left(x+\frac{11}{2}\right)^2=\frac{225}{4}\)

\(\Leftrightarrow\orbr{\begin{cases}x+\frac{11}{2}=\sqrt{\frac{225}{4}}\\x+\frac{11}{2}=-\sqrt{\frac{225}{4}}\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x+\frac{11}{2}=\frac{25}{2}\\x+\frac{11}{2}=-\frac{25}{2}\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=7\\x=-18\end{cases}}\)

phân tích mẫu thành nhân tử r áp dụng \(\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}\) sau đó rút gọn quy đồng

\(\frac{1}{x^2+9x+20}+\frac{1}{x^2+11x+30}+\frac{1}{x^2+13x+42}=\frac{1}{18}\) \(\left(ĐKXĐ:x\ne0;x\ne-4;x\ne-5;x\ne-6;x\ne-7\right)\)

\(\Leftrightarrow\frac{1}{x^2+4x+5x+20}+\frac{1}{x^2+5x+6x+30}+\frac{1}{x^2+6x+7x+42}=\frac{1}{18}\)

\(\Leftrightarrow\frac{1}{x\left(x+4\right)+5\left(x+4\right)}+\frac{1}{x\left(x+5\right)+6\left(x+5\right)}+\frac{1}{x\left(x+6\right)+7\left(x+6\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{\left(x+6\right)\left(x+7\right)+\left(x+4\right)\left(x+7\right)+\left(x+4\right)\left(x+5\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{\left(x^2+13x+42\right)+\left(x^2+11x+28\right)+\left(x^2+9x+20\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{x^2+13x+42+x^2+11x+28+x^2+9x+20}{\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{3x^2+33x+90}{\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\frac{3\left(x^2+11x+30\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Leftrightarrow\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)=18.3\left(x^2+11x+30\right)\)

\(\Leftrightarrow\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)=54\left(x+5\right)\left(x+6\right)\)

\(\Leftrightarrow\left(x+4\right)\left(x+7\right)=54\)

\(\Leftrightarrow x^2+11x+28-54=0\)

\(\Leftrightarrow x^2+11x-26=0\)

\(\Leftrightarrow x^2+13x-2x-26=0\)

\(\Leftrightarrow x\left(x+13\right)-2\left(x+13\right)=0\)

\(\Leftrightarrow\left(x+13\right)\left(x-2\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x+13=0\\x-2=0\end{cases}}\) 

\(\Leftrightarrow\orbr{\begin{cases}x=-13\left(tm\right)\\x=2\left(tm\right)\end{cases}}\)

rần Duy Thanh:gà dữ đến dấu <=> thứ 3 r` mà còn quy đồng. What is your favourite food :v

\(\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}\) 

\(\Leftrightarrow\frac{1}{\left(x+4\right)}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+6}+\frac{1}{x+6}-\frac{1}{x+7}\)

\(\Leftrightarrow\frac{1}{x+4}-\frac{1}{x+7}=\frac{1}{18}\)

\(\Leftrightarrow\frac{3}{\left(x+4\right)\left(x+7\right)}=\frac{1}{18}\)

\(\Rightarrow x^2+11x-26=0\)

\(\Rightarrow x=2\)

nếu x<0 thì có lẽ là x=-2

Ta có: \(\frac{1}{x^2+9x+20}\)\(+\frac{1}{x^2+11x+30}\)\(+\frac{1}{x^2+13x+42}\)

        =\(\frac{1}{x^2+4x+5x+20}\)\(+\frac{1}{x^2+5x+6x+30}\)\(+\frac{1}{x^2+6x+7x+42}\)

        =\(\frac{1}{x\left(x+4\right)+5\left(x+4\right)}\)\(+\frac{1}{x\left(x+5\right)+6\left(x+5\right)}\)\(+\frac{1}{x\left(x+6\right)+7\left(x+6\right)}\)

        =\(\frac{1}{\left(x+4\right)\left(x+5\right)}\)\(+\frac{1}{\left(x+5\right)\left(x+6\right)}\)\(+\frac{1}{\left(x+6\right)\left(x+7\right)}\)

        =\(\frac{1}{x+4}-\frac{1}{x+5}\)\(+\frac{1}{x+5}-\frac{1}{x+6}\)\(+\frac{1}{x+6}-\frac{1}{x+7}\)

        =\(\frac{1}{x+4}-\frac{1}{x+7}\)

        =\(\frac{x+7-x-4}{\left(x+4\right)\left(x+7\right)}\)=\(\frac{3}{\left(x+4\right)\left(x+7\right)}\)

\(\frac{1}{x^2+9x+20}\)  \(+\)  \(\frac{1}{x^2+11x+30}\)  \(+\)\(\frac{1}{x^2+13x+42}\)

= \(\frac{1}{\left(x+4\right)\left(x+5\right)}\)\(+\)  \(\frac{1}{\left(x+5\right)\left(x+6\right)}\)  \(+\) \(\frac{1}{\left(x+6\right)\left(x+7\right)}\)

= \(\frac{1}{x+4}\)\(-\)\(\frac{1}{x+5}\) \(+\)\(\frac{1}{x+5}\)\(-\)\(\frac{1}{x+6}\)\(+\)\(\frac{1}{x+6}\)\(-\)\(\frac{1}{x+7}\)

= \(\frac{1}{x+4}\)\(-\)\(\frac{1}{x+7}\)= \(\frac{x+7-\left(x+4\right)}{\left(x+4\right)\left(x+7\right)}\)= \(\frac{3}{x^2+11x+28}\)

Đáp án đây:

=\(\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+6\right)}+\frac{1}{\left(x+6\right)\left(x+7\right)}\)

=\(\frac{1}{x+4}-\frac{1}{x+7}\)

=\(\frac{3}{\left(x+4\right)\left(x+7\right)}\)

chắc thế thôi

bn giải nhanh nhỉ quang duy

giải giùm ra kết quả cho tui mừng coi

<=>1/(x+4)(x+5)+1/(x+5)(x+6)+1/(x+6)(x+7)=1/18

<=>1/(x+4)-1/(x+5)+1/(x+5)-1/(x+6)+1/(x+6)-1/(x+7)=1/18

<=>1/(x+4)-1/(x+7)=1/18

<=>3/(x+4)(x+7)=1/18

<=>(x+4)(x+7)=54

<=>x²+11x+28=54

<=>x²+11x-16=0

<=>x²+11x+121/4-185/4=0

đến đây chắctự làm đc

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