Find the equation of the equation with the root (α² - β²) = 3 and α - β = 2.

Steps to Solve

1. Define the Variables Let $\alpha$ and $\beta$ be the roots of the equation. We have two conditions: $$\alpha^2 - \beta^2 = 3$$ and $$\alpha - \beta = 2$$.

2. Express $\alpha$ in terms of $\beta$ From the second condition, we can rearrange the equation: $$\alpha = \beta + 2$$.

3. Substitute in the first condition Substituting $\alpha$ in the first equation: $$(\beta + 2)^2 - \beta^2 = 3$$.

4. Expand the equation Expanding the left side: $$\beta^2 + 4\beta + 4 - \beta^2 = 3$$. This simplifies to: $$4\beta + 4 = 3$$.

5. Solve for $\beta$ Rearranging gives us: $$4\beta = 3 - 4$$ which simplifies to: $$4\beta = -1$$, thus, $$\beta = -\frac{1}{4}$$.

6. Find $\alpha$ Now substituting $\beta$ back to find $\alpha$: $$\alpha = -\frac{1}{4} + 2 = \frac{7}{4}$$.

7. Form the Equation from Roots Using the roots $\alpha$ and $\beta$, the quadratic polynomial whose roots are given by the formula: $$x^2 - (\alpha + \beta)x + \alpha \beta = 0$$.

8. Calculate the Sum and Product of Roots

• Sum of the roots: $$\alpha + \beta = \frac{7}{4} - \frac{1}{4} = \frac{6}{4} = \frac{3}{2}$$.
• Product of the roots: $$\alpha \beta = \left(\frac{7}{4}\right) \left(-\frac{1}{4}\right) = -\frac{7}{16}$$.

9. Write the Equation Now, we can substitute into the standard form: $$x^2 - \left(\frac{3}{2}\right)x - \left(-\frac{7}{16}\right) = 0$$, which simplifies to: $$x^2 - \frac{3}{2}x + \frac{7}{16} = 0$$.