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

Steps to Solve

1. Identify the equations We have two conditions involving the roots $\alpha$ and $\beta$:

   • The difference of their squares: $$ \alpha^2 - \beta^2 = 3 $$
   • Their difference: $$ \alpha - \beta = 2 $$

2. Use the difference of squares formula The difference of squares can be factored as: $$ \alpha^2 - \beta^2 = (\alpha - \beta)(\alpha + \beta) $$ Substituting the value of $\alpha - \beta$: $$ 3 = 2(\alpha + \beta) $$

3. Solve for $\alpha + \beta$ Now, isolate $\alpha + \beta$: $$ \alpha + \beta = \frac{3}{2} $$

4. Set up the quadratic equation Now we have:

   • The sum of the roots: $s = \alpha + \beta = \frac{3}{2}$
   • The difference of the roots: $d = \alpha - \beta = 2$

Using the formulas for constructing a quadratic equation based on its roots, which is $x^2 - sx + p = 0$, we first need to find the product of the roots $\alpha \beta$.

5. Calculate $\alpha \beta$ We can use the equation: $$ \alpha = \beta + 2 $$ Substituting this in the sum: $$ \beta + 2 + \beta = \frac{3}{2} $$ $$ 2\beta + 2 = \frac{3}{2} $$ $$ 2\beta = \frac{3}{2} - 2 $$ $$ 2\beta = \frac{3}{2} - \frac{4}{2} = -\frac{1}{2} $$ $$ \beta = -\frac{1}{4} $$

Now we can find $\alpha$: $$ \alpha = \beta + 2 = -\frac{1}{4} + 2 = \frac{7}{4} $$

6. Determine the product of the roots Now, we calculate $\alpha \beta$: $$ \alpha \beta = \frac{7}{4} \cdot -\frac{1}{4} = -\frac{7}{16} $$

7. Final equation formulation Using the quadratic equation structure: $$ x^2 - \left(\frac{3}{2}\right)x - \left(-\frac{7}{16}\right) = 0 $$ This simplifies to: $$ x^2 - \frac{3}{2}x + \frac{7}{16} = 0 $$

8. Clear the fractions To eliminate the fractions, multiply through by 16: $$ 16x^2 - 24x + 7 = 0 $$