If the following quadratic equation has two equal and real roots,then find the value of $k$: $4x^{2} - 5x + k = 0$.

If one root of $x^{2}-5x+6=0$ is also the root of $x^{2}+3x+c=0$,then $c = \ldots$

State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has at most two roots.
$(ii)$ If the coefficient of $x^{2}$ and the constant term of a quadratic equation have opposite signs,then the quadratic equation has real roots.

Examine whether the following equation is quadratic or not: $3x^{2} + 5\sqrt{x} + 3 = 0$.

The hypotenuse of a garden in the shape of a right triangle is $1$ metre more than twice the shortest side. If the longer side containing the right angle is $7$ metres more than the shorter side,find the sides of the garden.

The quadratic equation having roots $-1$ and $2$ is .... .