2007 iTest Problems/Problem 36
Problem
Let b be a real number randomly selected from the interval . Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation has two distinct real solutions. Find the value of .
Solution
The equation has quadratic form, so complete the square to solve for x.
In order for the equation to have real solutions,
Note that is greater than or equal to when or . Also, if , then expression leads to and only has one unique solution, so discard as a solution. The rest of the values leads to equalling some positive value, so these values will lead to two distinct real solutions.
Therefore, in interval notation, , so the probability that the equation has at least two distinct real solutions when is randomly picked from interval is . This means that .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 35 |
Followed by: Problem 37 | |
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