Final answer:
The value of k in the quadratic function h(x) = 2x² + (k+4)x + k can be any real number. None of the given options is the correct value of k.
Step-by-step explanation:
The value of k in the quadratic function h(x) = 2x² + (k+4)x + k can be determined by comparing the given quadratic equation to the standard form ax² + bx + c = 0. In this case, we have a = 2, b = (k+4), and c = k. To find the value of k, we can use the fact that a quadratic equation has a real solution when its discriminant (b² - 4ac) is greater than or equal to zero. Plugging in the values, we get:
- The discriminant is (k+4)² - 4(2)(k).
- Setting the discriminant greater than or equal to zero, we have (k+4)² - 8k ≥ 0.
- Simplifying the equation, we get k² + 8k + 16 - 8k ≥ 0.
- Simplifying further, we get k² + 16 ≥ 0.
- Since k² is always greater than or equal to zero, the inequality is true for all values of k.
Therefore, the value of k can be any real number. None of the options A, B, C, or D is the correct value of k.