Solving Equations

 

Present your solutions to the following problems. Show your steps and
explain your reasoning clearly.
Problem 1 Find, if possible, all values of the constant a so that the the
vertex of the parabola y = −3x
2 + ax + 1 lies on the line y = 2x + 1. Do not
use any formula for the vertex of the parabola. Instead, complete the square
as we had done during our first lecture.
Problem 2 Solve for x if
2|x − 1| > 3|x + 1|.
Problem 3 Find the range of f, where f(x) = 3x
2−4x+3
.
1

 

Sample Solution

In order to test and determine whether an attempt at defining “good” is correct and not a concealed assignment is what Moore called the “open question argument.” Moore proposed that if “goodness” is a natural property, then there is some correct explanation of which natural property it is. For example, maybe “goodness” is the same property as “pleasantness”, or the same property as being “desirable”. Further, a correct property must be identified to fill in an identity statement of the form “goodness = __________”, or, “what is good is _________”. This kind of identity statement can be correct only if both terms on either side of the identity sign are synonyms for proficient speakers who understand both terms. Synonymy of the two terms is then tested through substitution of a term. Moore’s idea is that substitution of synonyms for one another preserves the original proposition that a sentence expresses. For example, using the sentence: “what is good is pleasant.” For this to pass Moore’s test, the sentence would have to express the same thing as “what is pleasant is pleasant.” Moore believed it was obvious that these two sentences do not express the same proposition. In thinking that what is good is pleasant, Moore thought one is not only thinking that what is pleasant is pleasant. According to Moore, there is an “open question” as to whether what is good is pleasant, and it can be understood when someone doubts the generated statement. However, there is no “open question” as to whether what is pleasant is pleasant, because this analytic truth cannot be doubted. Therefore, Moore thought that no substitution will pass the test. Therefore, there is no natural property of “goodness”. In other words, according to Moore and his open question argument, “goodness” is a non-natural property.

Objections to the open question argument include the fact that Moore

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