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

Find the roots of the following quadratic equation by using the quadratic formula,if they exist: $x^{2}-5x-1=0$

Find the discriminant of the following quadratic equation and hence determine the nature of the roots of the equation: $5x^{2} - 6x + 2 = 0$.

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

Which of the following groups correctly matches the data of Part $I$ with the data of Part $II$?

Part $I$	Part $II$
$1.$ The discriminant of $x^{2}+5x+6=0$	$a. 1$
$2.$ The discriminant of $x^{2}+5x+4=0$	$b. 9$
$3.$ The discriminant of $x^{2}+4x+3=0$	$c. 4$
$4.$ The discriminant of $x^{2}+6x+5=0$	$d. 16$

The difference between the lengths of the sides of a right triangle which form the right angle is $7 \, cm$. If its perimeter is $30 \, cm$,find the lengths of the sides.