If $3 + 4i$ is a root of the equation ${x^2} + px + q = 0$ where $p$ and $q$ are real numbers,then:

The real roots of the equation $x^2 + 5|x| + 4 = 0$ are

The positive root of $x^{2}-78.8=0$ after the first approximation by the Newton-Raphson method,assuming the initial approximation to the root is $14$,is:

The positive integer $n$, for which the solutions of the equation $x(x+2)+(x+2)(x+4)+...+(x+2n-2)(x+2n) = \frac{8n}{3}$ are two consecutive even integers, is:

If $m, n$ are the roots of the equation ${x^2} - x - 1 = 0$,then the value of $\frac{{\left( {1 + m{{\log }_e}3 + \frac{{{{(m{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)\left( {1 + n{{\log }_e}3 + \frac{{{{(n{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}{{\left( {1 + mn{{\log }_e}3 + \frac{{{{(mn{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}$ is:

If $|{x^2} - x - 6| = x + 2$,then the values of $x$ are