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### Probability That a Function Has Real Roots

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Date: 9/24/95 at 1:17:43
From: "ROBERT R. REICHEL"
Subject: help on probability

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
```
Associated Topics:
High School Probability

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