If the roots of the equation $ax^2 + bx + c = 0$ are $\alpha$ and $\beta$,then the value of $\alpha\beta^2 + \alpha^2\beta + \alpha\beta$ will be:

The number of real roots of the equation $\frac{P^2}{x} + \frac{Q^2}{x - 1} = 1$,where $P$ and $Q$ are non-zero real numbers,is

If $\alpha, \beta$ are the roots of the equation $x^{2}-5x+6=0$,construct a quadratic equation whose roots are $\frac{1}{\alpha}, \frac{1}{\beta}$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-16x+63=0$
$II.$ $y^{2}-2y-35=0$

The length of a rectangular plot is $8 \, m$ greater than its breadth. If the area of the plot is $308 \, m^{2}$,find the length of the plot.

Two students,while solving a quadratic equation in $x$,one copied the constant term incorrectly and got the roots $3$ and $2$. The other copied the constant term and coefficient of $x^2$ correctly as $-6$ and $1$ respectively. The correct roots are: