Final answer:
The solutions of the quadratic equation x^2 - 10x = -34 are x = 5 + 3i and x = 5 - 3i, which correspond to option B, when using the quadratic formula with complex numbers.
Step-by-step explanation:
We need to solve the quadratic equation x^2 - 10x = -34 by first bringing it to standard form, which means everything on one side of the equation set equal to zero. Thus, we add 34 to both sides:
x^2 - 10x + 34 = 0
To solve this quadratic equation, we use the quadratic formula:
x = ∛(-b ± √(b^2 - 4ac))/(2a)
Where a, b, and c are coefficients from the quadratic equation ax^2 + bx + c = 0. For our equation, a is 1, b is -10, and c is 34.
Substituting these values into the quadratic formula, we get:
x = (-(-10) ± √((-10)^2 - 4(1)(34)))/(2(1))
x = (10 ± √(100 - 136))/(2)
Since 100 - 136 is negative, we will have complex numbers as solutions:
x = (10 ± √(-36))/(2)
x = 5 ± 3i
Thus, the solutions to the equation are x = 5 + 3i and x = 5 - 3i, which corresponds to option B.