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?
 

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