If equation $ax^2+bx+c=0:;:(a,b,c \in R)$ and $2x^2+3x+4=0$ have a common root, then $a:b:c=2:3:4$

We find the discriminant of the equation 2x^2 + 3x + 4 = 0, which is 3^2 - 4(2)(4) = -23. Because the discriminant is negative, this equation has two non-real complex conjugate roots. If a real quadratic equation ax^2 + bx + c = 0 shares a root with it, the shared root must be non-real, meaning the two equations must share the exact same complex conjugate pair. This requires the equations to be identical up to a constant multiple, proving that the ratio a:b:c must be 2:3:4, making both the assertion and reason completely correct and directly linked.

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