That we may form an equation similar to the above, let us assume y = √a + √b + √c, and also suppose that the letters a, b, c denote the roots of the cubic equation
z³ + Pz² + Qz − R = 0;
then, from the theory of equations we have
a + b + c = −P, ab + ac + bc = Q, abc = R.
We square the assumed formula
y = √a + √b + √c,
and obtain y² = a + b + c + 2(√ab + √ac + √bc);
or, substituting −P for a + b + c, and transposing,
y² + P = 2(√ab + √ac + √bc).
Let this equation be also squared, and we have