Question

daca x1,x2 sunt solutiile ecuatiei x²-x+m=0,m∈R,aflati m stiind ca Ix1-x2I=1

Answer 1

$x^2-x+m = 0 \\ \\ x_1+x_2 = - \dfrac{b}{a} \Rightarrow x_1+x_2 =- \dfrac{-1}{1} \Rightarrow x_1+x_2 =1 \\ x_1\cdot x_2 = \dfrac{c}{a} \Rightarrow x_1\cdot x_1 = \frac{m}{1} \Rightarrow x_1\cdot x_2 = m \ \\ x_1^2+x_2^2 = (x_1+x_2)^2-2x_1x_2 \Rightarrow x_1^2+x_2^2 = 1 - 2m \\ \\ (x_1-x_2)^2 = x_1^2-2x_1x_2+x_2^2 \Rightarrow (x_1-x_2)^2 = x_1^2+x_2^2 - 2x_1x_2 \Rightarrow \\ \Rightarrow(x_1-x_2)^2 = 1-2m-2m \Rightarrow (x_1-x_2)^2 = 1-4m \Big| ^{ \dfrac{1}{2} } \Rightarrow$

$\Rightarrow \sqrt{(x_1-x_2)^2} = \sqrt{1-4m} \Rightarrow |x_1-x_2|=\sqrt{1-4m} \\ \\ \sqrt{1-4m} =1 \Rightarrow 1-4m = 1 \Rightarrow -4m = 0 \Rightarrow \boxed{m=0}$