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

If the product of the roots of the equation ${x^2} - 3kx + 2{e^{2\log k}} - 1 = 0$ is $7$,then for what value of $k$ will its roots be real?

Solve the given two equations and select the correct answer from the given options.
$I.$ $(289)^{\frac{1}{2}} x - \sqrt{324} = 203$
$II.$ $(484)^{\frac{1}{2}} y - \sqrt{225} = 183$

Let $[t]$ denote the greatest integer $\leq t$. Then the equation in $x$,$[x]^{2} + 2[x + 2] - 7 = 0$,has

If $p$ and $q$ are the roots of the equation ${x^2} + pq = (p + 1)x$,then $q=$

In the following,determine the set of values of $P$ for which the given quadratic equation $Px^2 + 4x + 1 = 0$ has real roots.