Prove that the graph of the function f(x) = 3*p^2*x^2 + 6px + 1 intersects the x-axis for all values of p.?

Answer 1

Use the discriminant test:
b^2 - 4ac = (6p)^2 - 4(3p^2)(1) = 36p^2 - 12p^2 = 24p^2

When the discriminant is non-negative, you have real roots (i.e. the function intersects the x-axis). Clearly this is positive for all values of p.

Answer 2

Essentially, all we have to do is prove that:

Delta>=0

(6p)^2 -4*3*p^2 = 24p^2

Since p^2 is always >= 0, QED