Question

Sa se determine m stiing ca solutiile x1 si x2 ale ecuatiei x²+(m+3)x+3=0 verifica x1=3x2
Ajutor va rog!;multumesc anticipat!

Answer 1

x1x2=x1*3x1=3x1²
3x1²=(Viete)=3/1  =3
x1²=1
x1=1   sau x1=-1

pt x1=1
1+m+3+3=0....m=-7

ptx1=-1
1-m-3+3=0....m=1
m∈{-7;1}

Answer 2

[tex]\it x^2+(m+3)x +3 = 0 \ \ \ \ (1) \\ \\ x_1=3x_2 \ \ \ \ (2)[/tex]

Din relațiile lui Viète rezultă:


$\it x_1x_2=3 \stackrel{(2)}{\Longrightarrow} 3x_2x_2=3|_{:3} \Rightarrow x_2^2=1 \Rightarrow \sqrt{x_2^2} =\sqrt1 \Rightarrow |x_2 | =1 \Rightarrow \\ \\ \Rightarrow x_2=\pm1 \\ \\ x_2=-1 \stackrel{(1)}{\Longrightarrow} (-1)^2+(m+3)\cdot(-1) +3=0 \Rightarrow \\ \\ \Rightarrow 1-m-3+3=0 \Rightarrow 1-m=0 \Rightarrow m=1$

[tex]\it x_2=1 \stackrel{(1)}{\Longrightarrow} 1^2+(m+3)\cdot1 +3=0 \Rightarrow \\ \\ \Rightarrow 1+m+3+3=0 \Rightarrow m +7=0 \Rightarrow m= -7 \\ \\ \\ Deci,\ m\in\{-7,\ 1\}[/tex]