Question

cho phương trình : x² -2(m+3)x +m² +3= 0

Tính A= x₁² + x₂² theo m

tìm m để x₁- x₂ = 6

Answer 1

\(\Delta'=\left(m+3\right)^2-m^2-3=6m+6\ge0\Rightarrow m\ge-1\)

Khi đó, pt có 2 nghiệm thỏa: \(\left\{{}\begin{matrix}x_1+x_2=2\left(m+3\right)\\x_1x_2=m^2+3\end{matrix}\right.\)

\(A=x_1^2+x_2^2=\left(x_1+x_2\right)^2-2x_1x_2=4\left(m+3\right)^2-2\left(m^2+3\right)\)

\(A=2m^2+24m+30\)

Ta có: \(\left\{{}\begin{matrix}x_1-x_2=6\\x_1+x_2=2m+6\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x_1=m+6\\x_2=m\end{matrix}\right.\)

Mà \(x_1x_2=m^2+3\Leftrightarrow m\left(m+6\right)=m^2+3\Rightarrow6m=3\Rightarrow m=\frac{1}{2}\)