MHB Using quadratic zeroes to find value of parameter

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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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