If $x = \sqrt{7 + 4\sqrt{3}},$ then $x + \frac{1}{x} = $

The roots of $4x^2 + 6px + 1 = 0$ are equal,then the value of $p$ is

Solve the given two equations and select the correct answer from the given options.
$I.$ $2x^2 + 13x - 7 = 0$
$II.$ $2y^2 - 5y + 3 = 0$

If $\alpha_1 < \alpha_2 < \alpha_3 < \alpha_4 < \alpha_5 < \alpha_6$,then the equation $(x - \alpha_1)(x - \alpha_3)(x - \alpha_5) + 3(x - \alpha_2)(x - \alpha_4)(x - \alpha_6) = 0$ has :-

If the roots of the equation $({p^2} + {q^2}){x^2} - 2q(p + r)x + ({q^2} + {r^2}) = 0$ are real and equal,then $p, q, r$ will be in

If the roots of the given equation $({m^2} + 1){x^2} + 2amx + {a^2} - {b^2} = 0$ are equal,then