Probability That a Function Has Real Roots
Date: 9/24/95 at 1:17:43 From: "ROBERT R. REICHEL" Subject: help on probability Please help! If r and s are numbers between 0 and 9, inclusive, what is the probability, P, that the function f(x) = x2 + rx + s has real roots? This question will consider only the case where r and s are integers. 1) calculate the probability the function f above has real roots for integers between 0 and 9, inclusive for both r and s. 2) calculate the probability the function f above has rational roots for integers between 0 and 9, inclusive for both r and s. Note: the number 2 in the function f(x) = x2 + rx + s is supposed to be a superscript impying x squared.
Date: 9/24/95 at 18:51:13 From: Doctor Andrew Subject: Re: help on probability I think a good way to approach this problem would be to use the quadratic equation, which states that for an equation ax^2 + bx +c = 0, x = [-b +/- sqrt(b^2 - 4ac)] / 2a Try to figure out which values of r and s will give you real and rational roots (answers to the above equations). You'll get real roots when the number in the square root is not negative, and you'll get rational roots when the number in the square root is a perfect square (0,1,4,9,etc.). You can do the probability part once you know these values; the probabilities can be found be figuring out what percent of all the possible values for r and s will give you real and rational roots. (In other words, how many pairs (r,s) give you rational or real roots divided by the number of total possible pairs (r,s) that you could plug into the equation.) The question ought to be more clear in stating that there is an equal probability for r and s having any of the values in the range. Otherwise, you're just left wondering what it means for a root to have a probability. I hope this helps. Give it your best shot, and if that doesn't seem to be enough get back to us with a more specific question about what's stumping you. Good luck! -Doctor Andrew, The Geometry Forum
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