The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if

If $|x - 2| + |x - 3| = 7$,then $x =$

The roots of the equation $a(x^2 + 1) - (a^2 + 1)x = 0$ are

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:

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 35x + 2 = 0$,then the value of $(2\alpha - 35)^3 \cdot (2\beta - 35)^3$ is equal to

Let $a, b \in R, a \neq 0$ be such that the equation $a x^{2}-2 b x+5=0$ has a repeated root $\alpha,$ which is also a root of the equation $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation,then $\alpha^{2}+\beta^{2}$ is equal to