Transforming Quadratic Equations: Finding Roots of Squared Polynomials

In this article, we will delve into the method of transforming a given quadratic equation to find a new equation whose roots are the squares of the original roots. This process involves understanding the relationships between the coefficients of the equation and the properties of the roots of polynomials. We'll explore a practical example where we start with the roots of a given equation and then determine the equation of the new roots.

Given Equation and Initial Calculations

The original quadratic equation is given by:

3x2 - 4x - 5 0

We are tasked with finding a new quadratic equation whose roots are the squares of the roots of the given equation. Let's denote the roots of the given equation by α and β. The first step is to calculate the sum and product of the roots using the formulas:

Sum of the roots: α β -b/a -(-4)/3 4/3

Product of the roots: αβ c/a -5/3

Calculating New Roots

The new roots of the desired equation are α2 and β2. To find these, we need to calculate the sum and product of the new roots based on the known sum and product of the original roots.

Sum of the New Roots

The sum of the new roots is given by:

α2 β2 (α - β)2 2αβ

Substituting the values we have:

α2 β2 (4/3)2 - 2(-5/3) 16/9 10/3

To combine the fractions, we convert 10/3 to a fraction with a denominator of 9:

10/3 30/9

Therefore:

α2 β2 16/9 30/9 46/9

Product of the New Roots

The product of the new roots is given by:

α2β2 (αβ)2 (-5/3)2 25/9

Forming the New Equation

Using the sum and product of the new roots, we can now form the quadratic equation with roots α2 and β2. The standard form of a quadratic equation with roots r1 and r2 is:

x2 - (r1 r2)x r1r2 0

Substituting the sum and product of the new roots, we get:

x2 - 46/9 x 25/9 0

To eliminate the fractions, we multiply the entire equation by 9:

9x2 - 46x 25 0

Final Result

The equation whose roots are α2 and β2 is:

boxed{9x2 - 46x 25 0}