For a quadratic equation,if $\ldots \ldots \ldots \ldots,$ then both the roots are equal.

$x = \dots$ is a solution of the quadratic equation $x^{2} + 7x + 12 = 0$.

Check whether the equation $6x^2 - 7x + 2 = 0$ has real roots, and if it has, find them by the method of completing the squares.

The denominator of a non-zero ratio is one less than twice the numerator. If the sum of the ratio and its reciprocal is $2 \frac{4}{15},$ find the ratio.

Find the discriminant of the following quadratic equation: $x^{2}-5x-14=0$.

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.