218. Without solving the quadratic equation $x^{2}+p x+q=0$, form a new equation whose roots are the squares of the roots of the given equation.
Solution. Let the roots of the given quadratic equation be: $x_{1}$ and $x_{2}$; the roots of the desired equation: $x_{1}^{2}$ and $x_{2}^{2}$. Then

$$
x_{1}^{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}=p^{2}-2 q ; \quad x_{1}^{2} \cdot x_{2}^{2}=q^{2}
$$

Thus, the desired equation will have the form:

$$
x^{2}-\left(p^{2}-2 q\right) x+q^{2}=0
$$