Transforming Quadratic Equations: Roots and Coefficients Transformation

When we have a quadratic equation, understanding and manipulating its roots can lead to some fascinating mathematical explorations. Let's delve into the process of transforming the roots of a quadratic equation into new ones, specifically focusing on the relationship between the roots (a) and (b). We begin with the equation (2x^2 - 6x - 5 0). This equation has two roots, (a) and (b), which lead us to explore the equation with roots (frac{a}{b}) and (frac{b}{a}).

Understanding the Original Equation

The given quadratic equation is:

2x^2 - 6x - 5 0

Using the standard form of a quadratic equation, we recognize that the sum of the roots (a b -frac{-6}{2} 3) and the product of the roots (ab frac{-5}{2}).

Formulating the New Equation

Given that we have the original roots (a) and (b) of the equation (2x^2 - 6x - 5 0), we want to find the equation with roots equal to (frac{a}{b}) and (frac{b}{a}).

We start by using the relationships:

ab frac{-5}{2}

ab -3

Next, we determine the product of the new roots:

(left(frac{a}{b}right) left(frac{b}{a}right) 1)

Now, let's calculate the sum of the new roots:

(left(frac{a}{b}right) left(frac{b}{a}right) frac{a^2 b^2}{ab} frac{(a b)^2 - 2ab}{ab})

Substituting the known values:

(frac{(a b)^2 - 2ab}{ab} frac{3^2 - 2left(frac{-5}{2}right)}{frac{-5}{2}} frac{9 5}{frac{-5}{2}} frac{14}{frac{-5}{2}} -frac{28}{5})

With the sum and product of the new roots, we can write the quadratic equation:

(x^2 - left(-frac{28}{5}right)x 1 0)

This simplifies to:

(5x^2 28x 5 0)

Generalizing the Approach

In a more general sense, if a quadratic equation (px^2 qx r 0) has roots that satisfy the same sum and product conditions, the process is as follows:

The sum of the roots (a b -frac{q}{p}) The product of the roots (ab frac{r}{p}) The new roots are (frac{a}{b}) and (frac{b}{a}) The product of the new roots is (left(frac{a}{b}right) left(frac{b}{a}right) 1) The sum of the new roots is (frac{(a b)^2 - 2ab}{ab})

Plugging these into the quadratic form:

(x^2 - left(frac{(a b)^2 - 2ab}{ab}right)x 1 0)

This transforms into a new quadratic equation, as demonstrated with the specific example provided.

Conclusion

Understanding the transformation of roots in quadratic equations can help us derive new equations with specific properties, like the one we have explored here. This process is not only mathematically intriguing but also a fundamental concept in the field of algebra and has various applications in physics, engineering, and other scientific disciplines.

Keywords: roots of quadratic equation, sum and product of roots, quadratic transformation