Final answer:
To find the standard form of a quadratic function with an a-value of 2 and a root of 5+2i, you first identify the conjugate root as 5-2i, create factors, expand them, and then multiply the quadratic equation by 2 to get 2t² - 20t + 58 = 0.
Step-by-step explanation:
The standard form of a quadratic function is ax² + bx + c = 0. To write a quadratic function with an a-value of 2 and a root of 5+2i, we must acknowledge that quadratic equations have conjugate roots when they are complex. Therefore, the other root must be 5-2i. Using these roots, we can create factors of the quadratic equation:
(t - (5+2i))(t - (5-2i)) = 0
Expanding this, we get:
t² - (5+2i)t - (5-2i)t + (5+2i)(5-2i) = 0
t² - 5t - 2it - 5t + 2it + 25 - (2i)(2i) = 0
t² - 10t + 25 + 4 = 0
t² - 10t + 29 = 0
Since the a-value must be 2, we multiply the entire equation by 2:
2t² - 20t + 58 = 0
This is the desired quadratic function in standard form.