Question

[Lizy]

`x^2 -(m+1)x+m=0` 

tìm m để pt có 2 nghiệm `x_1 , x_2` thỏa mãn \(x_1^2+x_2^2=\left(x_1-1\right)\left(x_2-1\right)-x_1-x_2+5\)

Answer 1

[Nguyễn Lê Phước Thịnh]

\(\text{Δ}=\left[-\left(m+1\right)\right]^2-4\cdot1\cdot m\)

\(=\left(m+1\right)^2-4m\)

\(=\left(m-1\right)^2>=0\forall m\)

=>Phương trình luôn có hai nghiệm

Theo Vi-et, ta có:

\(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{b}{a}=m+1\\x_1x_2=\dfrac{c}{a}=m\end{matrix}\right.\)

\(x_1^2+x_2^2=\left(x_1-1\right)\left(x_2-1\right)-x_1-x_2+5\)

=>\(\left(x_1+x_2\right)^2-2x_1x_2=x_1x_2-2\left(x_1+x_2\right)+6\)

=>\(\left(m+1\right)^2-2m=m-2\left(m+1\right)+6\)

=>\(m^2+1=m-2m-2+6\)

=>\(m^2+1=-m+4\)

=>\(m^2+m-3=0\)

=>\(m=\dfrac{-1\pm\sqrt{13}}{2}\)