Solve for a in the following equation using the quadratic formula:    

Graph the linear equation,

Suppose that a utility function u represents a preference relation on a set X. Consider the function that assigns to each x in X, f(x)=au(x)+b, where a and b are both constants. Suppose that the preference has at least a strict ranking (the agent is not indifference between at least one pair of alternatives). Then, f represents the preference if and only if,

Consider a preference relation on a finite set X={x1,x2,…,xn}. Suppose that it is complete and transitive. Then, the following function represents it: U(xi)= 3log|{j=1,…,n: xi at least as good as xj}|-2020, where |.| represents the cardinality of a set.

Luisa has preferences with the indifference map shown in the picture. Then,

Ana has preferences that are encoded by the Utility function U(x,y)=ax+y where a is a positive number. What is the value of a if Ana is indifferent between bundles A and B shown in the picture?

1) ________ is the assignment of value, or the amount a consumer must give to receive a product.

A box contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that the second ball is blue given that the first ball was red? Keep two digits.

A coastal surveillance system uses radar to detect incoming ships. The detection capability depends on weather conditions: Clear weather occurs 50% of the time, and the radar successfully detects ships with probability 0.95 Foggy weather occurs 30% of the time, and the radar successfully detects ships with probability 0.70 Stormy weather occurs 20% of the time, and the radar successfully detects ships with probability 0.50 What is the overall probability that an incoming ship is detected by the radar system? Keep three digits

In a manufacturing facility, a batch of 50 items contains 8 defective items. Quality control randomly selects 4 items from the batch to test. The batch is accepted if at most 1 of the tested items is defective. What is the probability that all 4 tested items are non-defective? (Use combinatorics) Keep two digits.