Solving Equations with Given Roots and Transformations

In this article, we will explore how to solve a quadratic equation given specific transformations of its roots. We will use a detailed step-by-step approach to find the equation with integral coefficients whose roots are α - 2 and β - 2, given the equation 2x2 - 4x - 1 0 and its roots α and β.

Understanding the Given Information

The original quadratic equation is given as 2x2 - 4x - 1 0. Here, we need to understand the significance of the equation and the roots α and β.

The equation can be written in standard form as:

[2x^2 - 4x - 1 0]

The roots of this quadratic equation are denoted by α and β. According to the theory of equations, the sum and product of the roots can be derived as follows:

Sum and Product of the Roots

Sum of the roots (α β): Using the formula -b/a, where b is the coefficient of the x-term and a is the coefficient of x2 term:

[α β -frac{-4}{2} 2]

Product of the roots (αβ): Using the formula c/a, where c is the constant term and a is the coefficient of x2 term:

[αβ frac{-1}{2} -frac{1}{2}]

Transforming the Roots

We need to find an equation whose roots are α - 2 and β - 2. Let's first determine the sum and product of these new roots.

Sum of the Transformed Roots

The sum of the transformed roots is:

[ (α - 2) (β - 2) α β - 4 2 - 4 -2 ]

Product of the Transformed Roots

The product of the transformed roots is:

[ (α - 2)(β - 2) αβ - 2α - 2β 4 ]

We know:

[ αβ -frac{1}{2} ]

[ α β 2 ]

Substituting these values:

[ (α - 2)(β - 2) -frac{1}{2} - 2(2) 4 -frac{1}{2} - 4 4 -frac{1}{2} ]

Forming the New Equation

Using the sum and product of the transformed roots, we can form the new quadratic equation. The general form of a quadratic equation is:

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

Substituting the values:

[ x^2 - (-2)x - frac{1}{2} 0 ]

Multiplying the entire equation by 2 to clear the fraction:

[ 2x^2 4x - 1 0 ]

This can be written as:

[ 2x^2 4x - 1 0 ]

Conclusion

We have successfully derived a new quadratic equation with integral coefficients that has the roots α - 2 and β - 2. The key steps involve understanding the sum and product of the original roots, transforming these values, and then using the transformation to form the new equation.

For further exploration, you can consider similar problems involving different transformations of roots to enhance your understanding of quadratic equations and their properties.