If α and β be the roots of x² + 7x + 12 = 0, find the quadratic equation whose roots are (α + β)² and (α - β)².

Given: α and β be the roots of x² + 7x + 12 = 0.

Then, sum of roots = - coefficient of x/coefficient of x²

^=> α + β = $\frac{- 7}{1}$ = -7 ------ (i)

The product of roots = $\frac{\mathrm{constant} \mathrm{term}}{\mathrm{coefficient} \mathrm{of} {\mathrm{x}}^2}$

=> αβ = $\frac{12}{1}$ = 12 ------(ii)

Quadratic equation with roots (α + β)² and (α - β)² is:

x² - [(α + β)² + (α - β)²]x + (α + β)²(α - β)² = 0   ---- (iii)

For the values of (α + β)² and (α - β)²:

=> (α + β)² = (-7)²

=>  (α + β)² = 49

Now, (α - β)² = (α + β)² - 4αβ

=> (α - β)² = (-7)² - 4(12) .....{from (i) & (ii)}

=> (α - β)² = 49 - 48

=> (α - β)² = 1

Putting all the values in (iii), we get:

=>  x² - [49 + 1]x + 49 ⋅ 1 = 0

=> x² - 50x + 49 = 0