Let $$[x]$$ denote the greatest integer function. What is the number of solutions of the equation $${x^2} - 4x + [x] = 0$$ in the interval [0, 2] ?

If p and q are the roots of the equation $${x^2} - 30x + 221 = 0$$, what is the value of p³ + q³?

Consider the following statements in respect of the quadratic equation $$4(x - p)(x - q) - {r^2} = 0$$, where p, q and r are real numbers.

1. The roots are real.

2. The roots are equal, if p = q and r = 0.

Which of the above statements is/are correct?

How many real roots does the equation $${x^2} + 3\left| x \right| + 2 = 0$$ have?