Final answer:
The standard form of the equation with p=6 and φ = 3π is x2−6x+9=0, as it is the only option that meets the sum and product of the roots as given by φ and p respectively.
Step-by-step explanation:
The question asks to find the standard form of the equation with p=6, φ = 3π. The standard form of a quadratic equation is ax2 + bx + c = 0. By comparing this with the given parameters p and φ, we can deduce that p is the product of the roots and φ is the sum of the roots. Therefore, we need to find the equation where the sum of the roots is -6 (since the sum is given as negative in the quadratic equation) and the product of the roots is 6.
Only option b) x2−6x+9=0 satisfies both conditions because the sum of the roots (which would be the opposite of -6 due to the standard form) is 6, and the product (which is the constant term) is 9. Since φ is 3π, which corresponds to a sum of roots of 6, and p=6 implies the product of roots should be 9, option b) is the correct equation.