Roots of Quadratic Equations
\begin{align}
 & \left( x-\alpha \right)\left( x-\beta \right)=0 \\ 
 & {{x}^{2}}-(\alpha +\beta )x+\alpha \beta =0 
\end{align}

\begin{align}
 & a{{x}^{2}}+bx+c=0 \\ 
 & {{x}^{2}}+\frac{b}{a}x+\frac{c}{a}=0 \\ 
\end{align}

Vieta’s Formulae:
Sum of Roots = $\alpha +\beta $ = $-\frac{b}{a}$ 

Product of Roots = $\alpha \beta =\frac{c}{a}$

Things to note:


\begin{align}
 & {{a}^{2}}+{{b}^{2}}={{(a+b)}^{2}}-2ab \\ 
 & {{a}^{3}}+{{b}^{3}}=(a+b)({{a}^{2}}-ab+{{b}^{2}}) \\ 
 & {{a}^{3}}-{{b}^{3}}=(a-b)({{a}^{2}}+ab+{{b}^{2}}) \\ 
 & {{a}^{4}}+{{b}^{4}}={{({{a}^{2}}+{{b}^{2}})}^{2}}-2{{a}^{2}}{{b}^{2}} \\ 
\end{align}

\begin{align}
  & \left( x-\alpha  \right)\left( x-\beta  \right)=0 \\ 
 & {{x}^{2}}-(\alpha +\beta )x+\alpha \beta =0  
\end{align}


\begin{align}
  & a{{x}^{2}}+bx+c=0 \\ 
 & {{x}^{2}}+\frac{b}{a}x+\frac{c}{a}=0 \\ 
\end{align}


Sum of Roots = $\alpha +\beta $ = $-\frac{b}{a}$  


Product of Roots = $\alpha \beta  =\frac{c}{a}$


\begin{align}
  & {{a}^{2}}+{{b}^{2}}={{(a+b)}^{2}}-2ab \\ 
 & {{a}^{3}}+{{b}^{3}}=(a+b)({{a}^{2}}-ab+{{b}^{2}}) \\ 
 & {{a}^{3}}-{{b}^{3}}=(a-b)({{a}^{2}}+ab+{{b}^{2}}) \\ 
 & {{a}^{4}}+{{b}^{4}}={{({{a}^{2}}+{{b}^{2}})}^{2}}-2{{a}^{2}}{{b}^{2}} \\ 
\end{align}

Sum and Products of Roots

Sum and Products of Roots