Equations

Concept:Use the identity $p^3+q^3 = (p+q)^3 - 3pq(p+q)$ to find the product of roots, then form the quadratic equation.Explanation:Let the roots be $p$ and $q$.Given: $p+q = 3$ and $p^3+q^3 = 7$.We know $p^3+q^3 = (p+q)^3 - 3pq(p+q)$.Substitute: $7 = 27 - 3pq \cdot 3 = 27 - 9pq$.So $9pq = 20$, hence $pq = \frac{20}{9}$.The quadratic equation is $x^2 - (p+q)x + pq = 0$, i.e., $x^2 - 3x + \frac{20}{9} = 0$.Multiply by $9$: $9x^2 - 27x + 20 = 0$.Among the options, only option A has the coefficients $9, -27, 20$, though it is written with $=20$ instead of $=0$; this is likely a typo. The correct equation is $9x^2 - 27x + 20 = 0$.Answer:Option A: $9x^2 - 27x + 20 = 20$ (interpreted as $9x^2 - 27x + 20 = 0$).