\(\alpha+\beta=\frac{-b}{a} ; \)
\(\alpha . \beta=\frac{c}{a}\)
\((\alpha+\beta)^{2}=\alpha^{2}+\beta^{2}-2 \alpha \cdot \beta \)
\(\left(\frac{-b}{a}\right)^{2}=\left(\alpha^{2}+\beta^{2}\right)-\frac{2 c}{a}\)
\(\therefore \alpha^{2}+\beta^{2}=\frac{b^{2}}{a^{2}}-\frac{2 c}{a}=\frac{b^{2}-2 c a}{a^{2}}\)