If the equation `x^(2) + 2z + 3 = 0 and ax^(2) + bx + c = 0 ` ,
a , b,c ` in` R, have a common root, then ` a: b: c: ` is

Options

A. `a + b omega + c omega^(2) = 0`
B. `a + b omega^(2) + c omega = 0`
C. `a^(3) + b^(3) + c^(3) = 3abc`
D. all the above

Answer

D

Explanation

Let `alpha` be the common root of the two equations. Then, `a alpha^(2) + b alpha + c = 0 and, b alpha^(2) + c alpha + a = 0`
Solving these two equations, we get `(alpha^(2))/(ab - c^(2))=(alpha)/(bc - a^(2))=(1)/(ac - b^(2))`
`rArr" "alpha^(2) = (ab - c^(2))/(ac - b^(2)) and alpha = (bc - a^(2))/(ac - b^(2))`
`(ab - c^(2))/(ac - b^(2)) = ((bc - a^(2))/(ac - b^(2)))^(2)" "[because alpha^(2) = (alpha)^(2)]`
`rArr" "(ab-c^(2))(ac-b^(2))=(bc-a^(2))^(2)`
`a(a^(3) + b^(3) + c^(3) - 3abc) = 0`
`rArr" "a^(3) + b^(3) + c^(3) - 3abc = 0" "[because a ne 0]`
`rArr" "(a+b+c) (a+b omega + c omega^(2))(a+ b omega^(2) + c omega)=0`
`rArr" "a + b omega + c omega^(2) = 0 or a + b omega^(2) + c omega = 0" "[beacuse a + b + c ne 0]`
NOTE To find the common root of two equations, we may make the coefficients of second degree terms in two equations equal and subtract. The value of x so obtained is the required common root.