Final answer:
To find the values of x such that the given vectors are orthogonal, set the dot product of the vectors equal to zero and solve for x using the quadratic formula.
Step-by-step explanation:
To find the values of x such that vectors ax + x, + 1 and b - 1, + 6x are orthogonal, we need to find the dot product of the two vectors and set it equal to zero. The dot product of two vectors is equal to the product of their corresponding components.
The dot product of ax + x, + 1 and b - 1, + 6x can be written as: (a * x) + (1 * -1) + (x * 6x) = 0
Simplifying the equation, we get: ax - x + 6x^2 = 0
Using the quadratic formula, we can solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values of a, b, and c from the equation, we have:
x = (1 ± √((-1)^2 - 4a(6))) / (2a)