Gọi hai nghiệm của A(x) là x1 và x2 và x1 = 2x2
Ta có: A(x1) = x12 - 5mx1 + 10m - 4 = 0
A(x2) = x22 - 5mx2 + 10m - 4 = 0
\(\Rightarrow\) x12 - 5mx1 + 10m - 4 = x22 - 5mx2 + 10m - 4 = 0
\(\Rightarrow\) x12 - 5mx1 - x22 + 5mx2 = 0
\(\Rightarrow\) 4x22 - 10mx2 - x22 + 5mx2 = 0
\(\Rightarrow\) 3x22 - 5mx2 = 0
\(\Rightarrow\) x2(3x2 - 5m) = 0
\(\Rightarrow\) x2 = 0 hoặc 3x2 - 5m = 0
\(\Rightarrow\) x2 = 0 hoặc 3x2 = 5m
\(\Rightarrow\) x2 = 0 hoặc x2 = \(\dfrac{5m}{3}\)
Với x2 = 0, ta có:
A(x2) = 0
\(\Leftrightarrow\) 10m - 4 = 0
\(\Leftrightarrow\) 10m = 4
\(\Leftrightarrow\) m = \(\dfrac{2}{5}\)
Với x2 = \(\dfrac{5m}{3}\), ta có:
A(x2) = 0
\(\Leftrightarrow\left(\dfrac{5m}{3}\right)^2-5m\left(\dfrac{5m}{3}\right)+10m-4=0\)
\(\Leftrightarrow\left(\dfrac{5m}{3}\right)^2-5m\left(\dfrac{5m}{3}\right)+10m=4\)
\(\Leftrightarrow\dfrac{25m^2}{9}-\dfrac{25m^2}{3}+10m-4=0\)
\(\Leftrightarrow25m^2-75m^2+90m-36=0\)
\(\Leftrightarrow50m^2-90m+36=0\)
\(\Leftrightarrow25m^2-45m+18=0\)
\(\Leftrightarrow25\left(m-\dfrac{6}{5}\right)\left(m-\dfrac{3}{5}\right)=0\)
\(\Leftrightarrow m-\dfrac{6}{5}=0\) hoặc \(m-\dfrac{3}{5}=0\)
\(\Leftrightarrow m=\dfrac{6}{5}\) hoặc \(m=\dfrac{3}{5}\)
Vậy \(m=\dfrac{2}{5}\), \(m=\dfrac{6}{5}\) hoặc \(m=\dfrac{3}{5}\)
