The quadratic equation whose roots are (p − q)/(p + q) and −(p + q)/(p − q) is:

Concept

For a quadratic equation whose roots are α and β, the monic form is x2 − (α + β)x + αβ = 0.

Multiplying the entire equation by any common non-zero factor does not change its roots; the given denominators must be non-zero.

Application

1. Let α = (p − q)/(p + q) and β = −(p + q)/(p − q).

2. Add the roots: α + β = [(p − q)2 − (p + q)2]/[(p + q)(p − q)] = −4pq/(p2 − q2).

3. Multiply the roots: αβ = [(p − q)/(p + q)] × [−(p + q)/(p − q)] = −1.

4. Substitute the sum and product into x2 − (α + β)x + αβ = 0 to obtain x2 + [4pq/(p2 − q2)]x − 1 = 0.

5. Multiply every term by p2 − q2 to obtain (p2 − q2)x2 + 4pqx − (p2 − q2) = 0, equivalently (p2 − q2)x2 + 4pqx − p2 + q2 = 0.

Cross-check

For (p2 − q2)x2 + 4pqx − p2 + q2 = 0, the coefficient relations give the root sum −4pq/(p2 − q2) and root product −1, exactly matching the two stated roots.

Therefore, the required equation is (p2 − q2)x2 + 4pqx − p2 + q2 = 0.