Consider the equation \[x^2-bx+c=0,\] where \(b\) and \(c\) are real numbers.

  1. Show that the roots of the equation are real and positive if and only if \(b>0\) and \(b^2\geq 4c>0\), and sketch the region of the \(b\)-\(c\) plane in which these conditions hold.

How can we show the roots of a quadratic equation are real? What more do we need to do to show that the roots are also positive?

What do we need to do to show an “if and only if” statement is true?