When are the roots for $x^2-bx+c=0$ real and positive?

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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.

2. Sketch the region of the \(b\)-\(c\) plane in which the roots of the equation are real and less than \(1\) in magnitude.