Given the quadratic equation x2 + px + q = 0 with roots alpha and beta, find the quadratic equation whose roots are (1)/(alpha2) and (1)/(beta2).

The given equation is $x^2 + px + q = 0$.
Let its roots be $\alpha$ and $\beta$.
Using Vieta's formulas:
Sum of roots: $\alpha + \beta = -p$
Product of roots: $\alpha \beta = q$

The new roots are $\frac{1}{\alpha^2}$ and $\frac{1}{\beta^2}$.
Let $S'$ be the sum of the new roots:

Combine the fractions:

We know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$.
Substitute the values from Step 1:

Also, $\alpha^2 \beta^2 = (\alpha\beta)^2 = q^2$.
Substitute these into the expression for $S'$:

Let $P'$ be the product of the new roots:

Substitute the value from Step 1:

A quadratic equation with roots $r_1$ and $r_2$ can be written as $x^2 - (r_1 + r_2)x + r_1 r_2 = 0$, or $x^2 - S'x + P' = 0$.
Substitute the calculated values for $S'$ and $P'$:

To eliminate the denominators, multiply the entire equation by $q^2$: