Using quadratic zeroes to find value of parameter

In summary, the conversation discusses finding the values of \(\alpha\) and \(\beta\) in terms of \(p\) and \(q\), using the equations \alpha^2-p\alpha+r=\frac{\alpha^2}{4}-q\frac{\alpha}{2}+r and \beta^2-p\beta+r=4\beta^2-2q\beta+r. The suggested approach is to solve for the non-zero values of \(\alpha\) and \(\beta\) using these equations.
  • #1
Mathsonfire
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Question 81
 
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  • #2
Re: Help

Do you have any thoughts on how to begin?
 
  • #3
Since there's been no reply, let's get started by observing that we must have:

\(\displaystyle r=\alpha\beta\)

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

\(\displaystyle \alpha^2-p\alpha+r=\frac{\alpha^2}{4}-q\frac{\alpha}{2}+r\)

\(\displaystyle \beta^2-p\beta+r=4\beta^2-2q\beta+r\)

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?
 

1. How do I find the value of the parameter using quadratic zeroes?

The value of the parameter can be found by using the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a. The values of a, b, and c can be obtained from the quadratic equation in the form of ax^2 + bx + c = 0.

2. Can I use any type of quadratic equation to find the value of the parameter?

Yes, the quadratic formula can be used to find the value of the parameter for any type of quadratic equation, as long as it is in the form of ax^2 + bx + c = 0.

3. Are there any other methods to find the value of the parameter besides using quadratic zeroes?

Yes, there are other methods such as factoring and completing the square that can also be used to find the value of the parameter. However, the quadratic formula is the most commonly used method.

4. Can I find the value of the parameter if the quadratic equation has imaginary solutions?

Yes, the quadratic formula can still be used to find the value of the parameter even if the solutions are imaginary. The only difference is that the value of the parameter will also be a complex number.

5. What is the significance of finding the value of the parameter using quadratic zeroes?

Finding the value of the parameter allows us to solve for the roots or solutions of the quadratic equation. This can help us determine the behavior and characteristics of the quadratic function, such as its minimum or maximum point, direction of opening, and the x-intercepts.

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