Final answer:
The missing term in the quadratic equation with the given root of 5 + (-3i) is 10, resulting in the equation x² - 10x + 34 = 0.
Step-by-step explanation:
To find the missing term in the quadratic equation x² - ____ + 34 = 0 given that one of its roots is 5 + (-3i), we can use the fact that complex roots of quadratic equations occur in conjugate pairs. This means if 5 - 3i is one root, the other root must be its conjugate 5 + 3i. By applying the Vieta's formulas for quadratic equations, which state that the sum of the roots is equal to the opposite of the coefficient of the x term divided by the coefficient of the x² term (which is 1 in this case), we can find the missing term.
The sum of the roots 5 - 3i and 5 + 3i is 10 (the imaginary parts cancel out). Therefore, the missing term in the equation is the negation of this sum, which is -10. Hence, the quadratic equation is x² - 10x + 34 = 0.