Which of the following statements are correct

            E₁) If a+b+c=0 then 1 is a root of $a{x}^2+bx+c=0$

            E₂) If $\sin \alpha , \cos \alpha$ are the roots of the equation $a{x}^2+bx+c=0$ then $b^2-a^2=2ac$

$$\begin{gathered} E_1: let f\left(x\right)=a{x}^2+bx+c \\ \Rightarrow f\left(1\right)=a+b+c \\ =0 \\ \therefore 1 is root of f\left(x\right) \end{gathered}$$

$$\begin{gathered} E_2: \\ \sin \alpha +\cos \alpha =\frac{-b}{a} , \sin \alpha \cos \alpha =\frac{c}{a} \\ \sin \alpha +\cos \alpha =\frac{-b}{a} \\ S.B.S \\ {( \sin \alpha +\cos \alpha )}^2={\left(\frac{-b}{a}\right)}^2 \Rightarrow {\sin }^2\alpha +{\cos }^2\alpha +2\sin \alpha \cos \alpha =\frac{b^2}{a^2} \\ \Rightarrow 1+2\left(\frac{c}{a}\right)=\frac{b^2}{a^2} \\ \Rightarrow a^2+2ac=b^2 \\ \Rightarrow b^2-a^2=2ac \end{gathered}$$