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

Question

If the equations $x^2+2x+3=0$ and $ax^2+bx+c=0$ ($a,b,c\in\mathbb R$) have a common root, then $a:b:c$ is

Accepted Answer

$x^2+2x+3=0$ has complex conjugate roots. Since the second equation has real coefficients and shares one root, it must share both --- so the equations are proportional: $a:b:c=1:2:3.$

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