If the equations x^2 + 2x + 3 = 0 and ax^2 + | vedclass

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(A) The given equation is $x^2 + 2x + 3 = 0$. Calculating the discriminant $D = b^2 - 4ac = (2)^2 - 4(1)(3) = 4 - 12 = -8$. Since $D < 0$,the roots of

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$1:2:3$

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(A) The given equation is $x^2 + 2x + 3 = 0$. Calculating the discriminant $D = b^2 - 4ac = (2)^2 - 4(1)(3) = 4 - 12 = -8$. Since $D If two quadratic equations have a common root and the coefficients are real,then the imaginary roots must occur in conjugate pairs. Therefore,if one root is common,both roots must be common. For two equations $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ to have the same roots,the ratio of their coefficients must be equal: $\frac{a}{1} = \frac{b}{2} = \frac{c}{3} = k$. This implies $a = k, b = 2k, c = 3k$. Thus,the ratio $a:b:c = k:2k:3k = 1:2:3$.

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

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