The common root of the equations $x^{2}-7x+10=0$ and $x^{2}-10x+16=0$ is:

The diagram shows the graph of $y = ax^2 + bx + c$. Then:

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+3ax+2a^{2}=0$ and if $\alpha^{2}+\beta^{2}=5$,find the value of $a$.

If the product of the roots of the equation $(a + 1)x^2 + (2a + 3)x + (3a + 4) = 0$ is $2$,then the sum of the roots is:

Number of integral values of $a$ for which both roots of the quadratic equation $x^2 - (2a + 3)x + a^2 + 3a = 0$ lie in the interval $(0, 4)$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $\frac{9}{\sqrt{x}} + \frac{19}{\sqrt{x}} = \sqrt{x}$
$II.$ $y^{5} - \frac{(2 \times 14)^{11/2}}{\sqrt{y}} = 0$