Final answer:
To determine the value of k for which the quadratic function x² - (k + 1)x + 3 has only one zero, we set the discriminant of the standard quadratic equation form ax²+bx+c = 0 equal to zero, leading to the solution k = 3, provided in option (c).
Step-by-step explanation:
To find the values of k for which the quadratic function x² - (k + 1)x + 3 has only one zero, we need to look for the condition where the discriminant (b² - 4ac) of the quadratic equation equals zero. This is because the discriminant tells us the nature of the roots: when it is zero, there is exactly one root.
The standard form of a quadratic equation is ax²+bx+c = 0. For our given function, a = 1, b = -(k + 1), and c = 3. We set the discriminant equal to zero and solve for k:
b² - 4ac = 0
⇒ (-(k + 1))² - 4(1)(3) = 0
⇒ (k + 1)² = 12
⇒ k² + 2k + 1 = 12
⇒ k² + 2k - 11 = 0
By factoring or using the quadratic formula, we can find the values of k that satisfy the equation. After solving, we find that k = 3 or k = -4. However, only k = 3 is listed among the answer choices provided, which is (c) k = 3.