Express through p and q the sum of the squares of the roots of the equation x2+px+q=0.

  p² -2q

Step-by-step explanation:

A quadratic with roots "a" and "b" can be written as ...

  (x -a)(x -b) = 0 = x² -(a+b)x +ab

Comparing this to the given equation, we find the relationship between p and q and the roots to be ...

  p = -(a+b)

  q = ab

If we square p we have a number that includes the sum of the squares of the roots:

  p² = (-(a+b))² = a² +2ab +b²

We can get rid of the 2ab term in this by making use of q:

  p² - 2q = a² +2ab + b² -2ab = a² +b² . . . . the desired sum