If equations ax^2+bx+c=0 (a,b,c R, a0) and 2x^2+3x+4=0 have a common root, then a:b:c equa…

Question

If equations $ax^2+bx+c=0$ ($a,b,c\in\mathbb R$, $a
e0$) and $2x^2+3x+4=0$ have a common root, then $a:b:c$ equals:

Accepted Answer

Since $2x^2+3x+4=0$ has non-real (complex conjugate) roots, and the two quadratics share a common root with real coefficients, they must share both roots --- hence be proportional: $a:b:c=2:3:4.$

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