cho a,b,c thỏa mãn a:2018=b:2019= c:2020. tính giá trị biểu thức M=(a-b)(b-c)-(c-a)^2

Đáp án:

$\begin{array}{l}
a:2018 = b:2019 = c:2020\\
 \Rightarrow \frac{a}{{2018}} = \frac{b}{{2019}} = \frac{c}{{2020}}\\
 = \frac{{a – b}}{{2018 – 2019}} = \frac{{b – c}}{{2019 – 2020}} = \frac{{c – a}}{{2020 – 2018}}\\
 = \frac{{a – b}}{{ – 1}} = \frac{{b – c}}{{ – 1}} = \frac{{c – a}}{2}\\
 \Rightarrow \left\{ \begin{array}{l}
a – b = b – c\\
c – a =  – 2\left( {a – b} \right)
\end{array} \right.\\
 \Rightarrow M = \left( {a – b} \right)\left( {b – c} \right) – {\left( {c – a} \right)^2}\\
M = \left( {a – b} \right).\left( {a – b} \right) – {\left( { – 2\left( {a – b} \right)} \right)^2}\\
 = {\left( {a – b} \right)^2} – 4{\left( {a – b} \right)^2}\\
 =  – 3{\left( {a – b} \right)^2}
\end{array}$