If $2+4i$ is one of the roots of $x^2+bx+c=0$ with $b, c \in R$,then $(b, c) =$

If ${x^2} + 2x + 2xy + my - 3$ has two rational factors,then the value of $m$ will be

Both the roots of the given equation $(x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0$ are always

The equation $4^{(x^2 + 2)} - 9 \cdot 2^{(x^2 + 2)} + 8 = 0$ has the solution:

The given image shows the graph of a quadratic equation $y = ax^2 + bx + c$. Based on the graph,which of the following is true?

Let $S = \{ \alpha : \log_2(9^{2\alpha-4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha-4} + 1) = 2 \}$. Then the maximum value of $\beta$ for which the equation $x^2 - 2(\sum_{\alpha \in S} \alpha)^2 x + \sum_{\alpha \in S} (\alpha+1)^2 \beta = 0$ has real roots,is $...........$