The Probability Both Roots are Positive and Real
This problem is from Gary Gordon, Lafayette College
Let f(x) = x2 + bx + c where b and c are real numbers between -1 and 1. What is the probability that both roots of f(x) are positive real numbers?
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Solution to The Probability Both Roots are Positive and Real
Students submitting correct solutions: Wyatt Witemeyer,
Let r and s be the roots of f(x), and consider the unit square in the plane with the horizontal axis representing (say) b values and the vertical axis representing c values.
Since f(x) = (x – r)(x – s) = x2 – (r+s)x + rs, it follows that b must be negative and c must be positive for both r and s to be positive. This happens in the top left quadrant of the unit square.
For the roots to both be real, we must have b2 larger than or equal to 4c. The region that satisfies this condition is below the graph of the curve given by c = b2 / 4.
By integration of the function x2 / 4, we find the area of the region satisfying both conditions above to be 1/12. Since the total area of the unit square is 4, the probability that both roots are positive and real is
1/48.
