Final answer:
The zeros of the function h(p) = p² - 5p - 5 are found using the quadratic formula and are (5 + 3√5) / 2 and (5 - 3√5) / 2. These are the real zeros of the function, with no non-real solutions.
Step-by-step explanation:
Finding Zeros Using the Quadratic Formula
To find all zeros of the function h(p) = p² - 5p - 5, we will use the quadratic formula. The general form of a quadratic equation is ax² + bx + c = 0. In this case, we compare h(p) with the general form and identify a = 1, b = -5, and c = -5.
The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a). Plugging in our values:
x = (-(-5) ± √((-5)² - 4(1)(-5))) / (2(1))
x = (5 ± √(25 + 20)) / 2
x = (5 ± √45) / 2
We simplify √45 to 3√5. Therefore, the zeros of the function are:
p = (5 + 3√5) / 2
p = (5 - 3√5) / 2
These are the real zeros since both are real numbers. There are no non-real zeros because the discriminant (b² - 4ac) is positive, indicating that both solutions are real numbers.