Deriving a Quadratic Equation from Given Roots: A Step-by-Step Guide

In this article, we'll walk through the process of finding a quadratic equation whose roots are the squares of the original roots of a given quadratic equation. This is a valuable exercise for understanding the relationships between the roots of a quadratic equation. We'll start with an example and guide you through the calculations.

Original Equation and Roots

Consider the quadratic equation:

3x^2 - 5x - 7 0

The roots of this equation are denoted as A and B. We need to find the new quadratic equation whose roots are A^2 and B^2. To do this, we'll use the relationships between the roots of the quadratic equation and the coefficients of the equation.

Understanding the Relationships

For any quadratic equation in the form ax^2 bx c 0:

The sum of the roots (A B) is given by -b/a. The product of the roots (AB) is given by c/a.

Given the original equation 3x^2 - 5x - 7 0:

The sum of the roots: α β -(-5)/3 5/3 The product of the roots: αβ -7/3

Deriving the New Roots

Next, we need to find the sum and product of the new roots, α^2 and β^2.

Sum of the New Roots

The sum of the new roots α^2 β^2 can be computed using the identity:

α^2 β^2 (α β)^2 - 2αβ

Substituting the values we found:

α^2 β^2 (5/3)^2 - 2(-7/3) 25/9 14/3

To combine these, we convert 14/3 to a fraction with a denominator of 9:

14/3 42/9

Thus:

α^2 β^2 25/9 - 42/9 -17/9

Product of the New Roots

The product of the new roots α^2β^2 is given by:

α^2β^2 (αβ)^2

Substituting the values:

α^2β^2 (7/3)^2 49/9

Forming the New Quadratic Equation

Using the sum and product of the new roots, we can form the new quadratic equation using the standard form:

x^2 - (sum of new roots)x (product of new roots) 0

Substituting the values we computed:

x^2 - (-17/9)x 49/9 0

This simplifies to:

x^2 17/9x 49/9 0

To clear the fractions, we multiply through by 9:

9x^2 17x 49 0

Thus, the equation whose roots are A^2 and B^2 is:

boxed{9x^2 17x 49 0}