Solving for a New Quadratic Equation with Transformed Roots from Given Roots

Understanding the transformation of roots in quadratic equations opens up a fascinating exploration in algebra. One such problem involves starting with the roots of a given quadratic equation and deriving a new quadratic equation from a transformed version of these roots. In this article, we will walk through the detailed steps to solve such a problem.

Given Problem and Solution Approach

The problem at hand involves finding the quadratic equation with the roots transformed as α2 and β2, where α and β are the roots of the quadratic equation x2 - 2x 3 0.

Step 1: Finding the Original Roots

To start, we need to find the roots of the original equation x2 - 2x 3 0. We can use the quadratic formula:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Here, a 1, b -2, and c 3. Substituting these values into the quadratic formula, we get:

[ x frac{2 pm sqrt{(-2)^2 - 4 cdot 1 cdot 3}}{2 cdot 1} ]

[ x frac{2 pm sqrt{4 - 12}}{2} ]

[ x frac{2 pm sqrt{-8}}{2} ]

[ x frac{2 pm 2isqrt{2}}{2} ]

[ x 1 pm isqrt{2} ]

Thus, the roots are:

[ alpha 1 isqrt{2}, beta 1 - isqrt{2} ]

Step 2: Transformed Roots

The next step involves transforming these roots as α2 and β2. We calculate these values as follows:

[ alpha^2 (1 isqrt{2})^2 1 2isqrt{2} - 2 -1 2isqrt{2} ]

[ beta^2 (1 - isqrt{2})^2 1 - 2isqrt{2} - 2 -1 - 2isqrt{2} ]

Step 3: Sum and Product of Transformed Roots

Using the properties of the roots, we can calculate the sum and the product of the transformed roots:

[ text{Sum} alpha^2 beta^2 (-1 2isqrt{2}) (-1 - 2isqrt{2}) -2 ]

[ text{Product} alpha^2 cdot beta^2 (-1 2isqrt{2})(-1 - 2isqrt{2}) (-1)^2 - (2isqrt{2})^2 ]

[ 1 - 4(-2) 1 8 9 ]

Step 4: Constructing the New Quadratic Equation

Now, using the sum and product of the transformed roots, we can form the new quadratic equation:

[ x^2 - (text{sum of roots})x (text{product of roots}) 0 ]

[ x^2 - (-2)x 9 0 ]

[ x^2 2x 9 0 ]

Conclusion

The new quadratic equation with the roots transformed as α2 and β2 from the roots of the equation x2 - 2x 3 0 is:

[ boxed{x^2 2x 9 0} ]

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