MHB -aux.15.IB23.family of functions .calculate the probability

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The discussion centers on calculating the probability of a quadratic function crossing the x-axis based on the family of functions defined by the equation x^2 + 3x + k. By completing the square, it is determined that the vertex is below the x-axis, indicating that the function crosses the x-axis only for k values of 1 and 2. This leads to a calculated probability of 2/7 for the function crossing the x-axis. Additionally, the discriminant method is mentioned as a more straightforward approach to find when there are two distinct real roots, confirming the results for k = 1 and k = 2. Overall, both methods provide insight into the behavior of the quadratic function in relation to the x-axis.
karush
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well by completing the square from $x^2+3x$ we get $\pmatrix {x+\frac{3}{2}}^2 -\frac{9}{4}$
which means the vertex is more than $2$ below the x-axis but above at $3$

so from the set k only at $k=1$ and $k=2$ will it cross the x-axis so $\frac{2}{7}$ is the probability that the function will cross the x-axis.

one way i guess
 
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You could also use the discriminant of f(x).

$3^2 - 4k > 0$

Works only for k = 1 . k = 2

:)
 
While completing the square certainly works, I agree with agentmulder's suggestion of using the discriminant as a means to determine when there will be two distinct real roots. It seems more straightforward to me.
 
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