$\begin{align}
& \text{Equation (i)} \\
& a{{x}^{2}}+bx+c=0\text{ whose roots are }\alpha \text{ and }\beta \\
& \text{Sum of roots }\alpha +\beta =-\frac{b}{a} \\
& \text{Product of roots }\alpha \beta =\frac{c}{a} \\
& \text{Equation (ii)} \\
& a'{{x}^{2}}+b'x+c'=0\text{ whose roots are }\zeta \text{ and }\delta \\
& \text{Sum of roots}=\zeta +\delta =-\frac{b'}{a'} \\
& \text{Product of roots}=\zeta \delta =\frac{c'}{a'} \\
& \text{For the new equation to be formed} \\
& \text{Sum of new roots}=\left( \frac{\alpha }{\zeta }+\frac{\beta }{\delta } \right)+\left( \frac{\alpha }{\delta }+\frac{\beta }{\zeta } \right) \\
& =\frac{\alpha \delta +\zeta \beta +\alpha \zeta +\zeta \beta }{\zeta \delta }=\frac{\delta (\alpha +\beta )+\zeta (\alpha +\beta )}{\zeta \delta }=\frac{(\alpha +\beta )(\zeta +\delta )}{\zeta \delta }=\frac{\left( -\frac{b}{a} \right)\left( -\frac{b'}{a'} \right)}{\left( \frac{c'}{a'} \right)}=\frac{bb'}{ac'} \\
& \text{Sum of roots}=\frac{bb'}{ac'} \\
& \text{Product of roots}=\left( \frac{\alpha }{\zeta }+\frac{\beta }{\delta } \right)\left( \frac{\alpha }{\delta }+\frac{\beta }{\zeta } \right)=\left( \frac{\alpha \delta +\zeta \beta }{\zeta \delta } \right)\left( \frac{\alpha \zeta +\beta \delta }{\zeta \delta } \right) \\
& =\frac{{{\alpha }^{2}}\delta \zeta +\alpha {{\delta }^{2}}\beta +\alpha \beta {{\zeta }^{2}}+{{\beta }^{2}}\delta \zeta }{{{(\zeta \delta )}^{2}}}=\frac{({{\alpha }^{2}}\delta \zeta +{{\beta }^{2}}\delta \zeta )+(\alpha {{\delta }^{2}}\beta +\alpha \beta {{\zeta }^{2}})}{{{(\zeta \delta )}^{2}}} \\
& =\frac{(\zeta \delta )({{\alpha }^{2}}+{{\beta }^{2}})+(\alpha \beta )({{\zeta }^{2}}+{{\delta }^{2}})}{{{(\zeta \delta )}^{2}}}=\frac{(\zeta \delta )\left[ {{(\alpha +\beta )}^{2}}-2\alpha \beta \right]+(\alpha \beta )\left[ {{(\zeta +\delta )}^{2}}-2\zeta \delta \right]}{{{(\zeta \delta )}^{2}}} \\
& =\frac{\left( \tfrac{c'}{a'} \right)\left[ {{\left( -\tfrac{b}{a} \right)}^{2}}-2\left( \tfrac{c}{a} \right) \right]+\left( \tfrac{c}{a} \right)\left[ {{\left( -\tfrac{{{b}^{1}}}{{{a}^{1}}} \right)}^{2}}-2\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right) \right]}{{{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)}^{2}}} \\
& =\frac{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)\left[ \tfrac{{{b}^{2}}}{{{a}^{2}}}-\tfrac{2c}{a} \right]+\left( \tfrac{c}{a} \right)\left[ \tfrac{{{({{b}^{1}})}^{2}}}{{{({{a}^{1}})}^{2}}}-\tfrac{2{{c}^{1}}}{{{a}^{1}}} \right]}{{{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)}^{2}}}=\frac{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)\left[ \tfrac{{{b}^{2}}-2ac}{{{a}^{2}}} \right]+\left( \tfrac{c}{a} \right)\left[ \tfrac{{{({{b}^{1}})}^{2}}-2{{a}^{1}}{{c}^{1}}}{{{({{a}^{1}})}^{2}}} \right]}{{{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)}^{2}}} \\
& \text{The new equation will be } \\
^2-\frac{b{{b}^{1}}}{a{{c}^{1}}}x+\frac{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)\left[ \tfrac{{{b}^{2}}-2ac}{{{a}^{2}}} \right]+\left( \tfrac{c}{a} \right)\left[ \tfrac{{{({{b}^{1}})}^{2}}-2{{a}^{1}}{{c}^{1}}}{{{({{a}^{1}})}^{2}}} \right]}{{{\left( \tfrac{{{c}^{1}}}{{{a}^{1}}} \right)}^{2}}}=0 \\
\end{align}$