1. the first formula found for solving cubic equations applied only to cubics without a square term. in modern notation, these look like these are sometimes called "depressed cubics." let y x- and substitute in the equation below to "red uce" the cubic. verify that the square term will disappear. now we can solve this equation for x using the formula forx, and then subtra ind y

Asays "we are both knaves" and b says nothing. exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by smullyan [sm78]) who can either lie or tell the truth. you encounter three people, a, b, and c. you know one of these people is a knight, one is a knave, and one is a spy. each of the three people knows the type of person each of other two is. for each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. when there is no unique solution, list all possible solutions or state that there are no solutions. 24. a says "c is the knave," b says, "a is the knight," and c says "i am the spy."

This is so hard and it’s for a test

Liz had 140 pens and inna had 100 pens. after inna gave liz some pens, liz had 3 times as many pens as inna. how many pens did inna give liz?