If two equations ` x^(2) + ax + bc = 0 and x^(2) + bx + ca =0` have a common root, the find the condition and the quadratic with other roots of the equations.

Explanation

Let ` alpha , beta` be the roots ` of x^(2) ,+ ax + bc=0 and alpha , gamma` be that of ` x^(2) + bx + ca =0` so that ` alpha` .is a commom root.
Then ` alpha^(2) + aalpha + bc=0 " alpha + beta = -a , alpha beta,=bc`
` alpha^(2) + balpha + ca=0, alpha + gamma , alpha gamma = ca`
Solving by cross- multiplication ,we have
`alpha^(2)/(a^(2)c-b^(2)c) = alpha/(bc-ca)= 1/(b-a)`
` Rightarrow alpha^(2)/(c(a-b)(a+b)) = alpha/(c(b-a)) = 1/(b-a)`
`Rightarrow alpha = -(a +b) =c`
` Rightarrow a + b + c=0`
Which is the required condition for a common root between given equations.
also ` alphabeta = bc = cbeta and alphagamma = ca = cgamma`
` Rightarrow beta = b Rightarrow gamma =a`
Thus the quadratic equation with roots ` beta and gamma`
is `x^(2) = (beta+ gamma) x + betagamma=0`
` Rightarrow x^(2)- (a +b) x + ab =0`
` Rightarrow x^(2) + cx + ab=0`