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Mathsonfire

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Question 81

Question 81

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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.

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Mathsonfire

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Question 81

Question 81

Mathematics news on Phys.org

- #2

MarkFL

Gold Member

MHB

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Do you have any thoughts on how to begin?

- #3

MarkFL

Gold Member

MHB

- 13,288

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

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?

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**.

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**.

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.

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.

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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