Find the quadratic equation with the root (α²,B) = 3 and α - β = 2.

Steps to Solve

1. Identify the given information

We have a root $(\alpha^2, B) = 3$, which means $\alpha^2 + B = 3$. Additionally, we know that $\alpha - \beta = 2$.

2. Express B in terms of α

From the first equation, we can isolate $B$: $$ B = 3 - \alpha^2 $$

3. Express β in terms of α

Using the second equation, we can express $\beta$ in terms of $\alpha$: $$ \beta = \alpha - 2 $$

4. Sum and Product of Roots

The roots of a quadratic equation can be denoted as $\alpha$ and $\beta$. The sum of the roots is given by: $$ \text{Sum} = \alpha + \beta $$ Replacing $\beta$: $$ \text{Sum} = \alpha + (\alpha - 2) = 2\alpha - 2 $$

The product of the roots is given by: $$ \text{Product} = \alpha \cdot \beta = \alpha(\alpha - 2) = \alpha^2 - 2\alpha $$

5. Form the quadratic equation

Using the standard form of a quadratic equation $x^2 - (\text{Sum})x + \text{Product} = 0$: $$ x^2 - (2\alpha - 2)x + (\alpha^2 - 2\alpha) = 0 $$

6. Final simplification

Simplifying the quadratic equation yields: $$ x^2 - (2\alpha - 2)x + (\alpha^2 - 2\alpha) = 0 $$

This is the quadratic equation formed using the roots based on the given conditions.