MHB Using quadratic zeroes to find value of parameter

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The discussion focuses on using quadratic equations to determine the values of parameters \(\alpha\) and \(\beta\) based on their relationships with \(p\) and \(q\). The equations presented indicate that the product of the roots \(r\) is equal to \(\alpha\beta\). Participants are encouraged to manipulate the quadratic equations to isolate \(\alpha\) and \(\beta\) in terms of the given parameters. The goal is to find non-zero solutions for both variables. This approach aims to clarify the connections between the parameters and the roots of the quadratic equations.
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Question 81
 
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Re: Help

Do you have any thoughts on how to begin?
 
Since there's been no reply, let's get started by observing that we must have:

$$r=\alpha\beta$$

Now, to express \(\alpha\) and \(\beta\) in terms of \(p\) and \(q\) I would look at:

$$\alpha^2-p\alpha+r=\frac{\alpha^2}{4}-q\frac{\alpha}{2}+r$$

$$\beta^2-p\beta+r=4\beta^2-2q\beta+r$$

Solve the first equation for the non-zero value of \(\alpha\) and solve the second equation for the non-zero value of \(\beta\)...what do you find?
 

Thread 'Significant figures for inherently bound values'

I have been insisting to my statistics students that for probabilities, the rule is the number of significant figures is the number of digits past the leading zeros or leading nines. For example to give 4 significant figures for a probability: 0.000001234 and 0.99999991234 are the correct number of decimal places. That way the complementary probability can also be given to the same significant figures ( 0.999998766 and 0.00000008766 respectively). More generally if you have a value that...
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