The name of the Indian mathematician who gave the formula for finding the roots of the quadratic equation $ax^2 + bx + c = 0$ by the method of completing the square is ..... .

Solve the following equation using the method of factorization: $\frac{x^{2}-1}{x^{2}+1}=\frac{7}{9}$

Solve the following equation using the quadratic formula,if the equation has a solution in $R$: $5x^{2} - 3x - 2 = 0$.

For a quadratic equation,if $\ldots \ldots \ldots \ldots$ then both the roots of an equation are distinct and real.

$A$ train travels at a certain average speed for a distance of $63 \, km$ and then travels a distance of $72 \, km$ at an average speed of $6 \, km/h$ more than its original speed. If it takes $3 \, hours$ to complete the total journey,what is its original average speed in $km/h$?

State whether the quadratic equation $\sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x + \frac{1}{\sqrt{2}} = 0$ has two distinct real roots. Justify your answer.