If the ratio of the roots of the equation `x^2+p x+q=0` are equal to ratio of the roots of the equation `x^2+b x+c=0` , then prove that `p^(2)c=b^2qdot`

Explanation

`alpha + beta = - p, alpha beta = q` (1)
`gamma + delta = - b, gamma delta = c` (2) we have ,
`(alpha)/(beta) = (gamma)/(delta) or (alpha + beta)/(alpha - beta) = (gamma + delta)/(gamma-delta)` [Using componendo and dividendo]
or `((alpha-beta)^(2))/((alpha + beta)^(2))=((gamma - delta)^(2))/((gamma+delta)^(2))`
or `((alpha-beta)^(2)- 4alpha beta)/((alpha + delta))=((gamma - delta)^(2) - 4gammadelta)/((gamma+delta)^(2))`
or ` -1(4alphabeta)/(alpha +beta)^(2)=1-(4gamma - delta)/((gamma+delta)^(2))`
or `(alpha beta)/((alpha + beta)^(2))=(gammadelta)/(gamma + delta)^(2)`
or `(q)/(p^(2) )=(c)\/(b^(2))`
or `p^(2) c = b^(2) q`