If $\sqrt{3x^2 - 7x - 30} + \sqrt{2x^2 - 7x - 5} = x + 5$,then $x$ is equal to

The roots of the equation $a(b-c)x^2 + b(c-a)x + c(a-b) = 0$ are

Let $f(x)$ be a quadratic polynomial such that $f(-1)+f(2)=0$. If one of the roots of $f(x)=0$ is $3$,then its other root lies in

For what value of $k$ will the equation $x^2 - (3k - 1)x + 2k^2 + 2k = 0$ have equal roots?

Let $S = \{x \in \mathbb{R} : (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10\}$. Then the number of elements in $S$ is:

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 $...........$