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, Answer Key) | ||
Preceded by: Problem 35 |
Followed by: Problem 37 | |
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