<p>pg. 684</p>
<p>Let [x] be defined as [x] = x^2 - x
for all values of x. if [a] = [a-2]
what is the value of a?</p>
<p>A 1
B 1/2
C 3/2
D 6/5
E 3</p>
<p>The answer is C, but I have no idea how to approach this question</p>
<p>This is one of those where you have to figure out what’s going on based on the example. This seems like a straightforward one as well. If = x^2 - x then [a] = [a-2] would work out to be a^2-a= (a-2)^2-a+2. Once you solve for a you eventually get 4a=6 and then a=3/2.</p>
<p>They’re just defining a new operation here. Don’t let it throw you off. Tackle the problem piece by piece.</p>
<p>What is [a]?
[a] = a^2 - a</p>
<p>What is [a-2]?
[a-2] = (a-2)^2 - (a-2)
= a^2 - 4a + 4 - a + 2
= a^2 - 5a + 6</p>
<p>[a] = [a-2]
a^2 - a = a^2 - 5a + 6
4a = 6
a = 6/4 = 3/2</p>
<p>Almost any question that asks for the value of a single variable (e.g., “what is y?” “then x=” “for what value of a is…”) and has numbers in the answer choices can be solved by throwing in those numbers for the variable. In other words, you can “backtrack.”</p>
<p>On this particular question, though, if you’re going to throw in the answer choice numbers, you would nevertheless need to understand what jamesford referred to as a “new operation” (or what I refer to as a “weird symbol function”).</p>
<p>These “weird symbol functions” act almost exactly the same as *f<a href=“%5BI%5Dx%5B/I%5D”>/I</a> type functions. So, on this question, = x^2 - x can be thought of as *f<a href=“%5BI%5Dx%5B/I%5D”>/I</a> = x^2 - x. From that point, you just need to remember that whatever value you put in parentheses after the *f<a href=“in%20place%20of%20the%20%5BI%5Dx%5B/I%5D”>/I</a> is what needs to be substituted for x on the right side of the equals sign. For instance, *f<a href=“4”>/I</a> would be 4^2 - 4 = 12.</p>
<p>If you understand all of the above, then you could simply start plugging in the answer choices to see which value for a will make [a] = <a href=“or%20%5BI%5Df%5B/I%5D(%5BI%5Da%5B/I%5D)%20=%20%5BI%5Df%5B/I%5D(%5BI%5Da%5B/I%5D%20-%202)”>a-2</a>.</p>
<p>Now, because numeric answer choices on the SAT are almost always given in ascending or descending order, it makes sense to start right in the middle at answer choice (C). Doing so is often more efficient because, if (C) turns out not to work, you may be able to tell whether you need to try a larger or smaller number and then move to the appropriate answer, thereby avoiding the need to try the other 2 answer choices (e.g, if you see you need to move from (C) to (B), you may not need to check (D) and (E)…assuming you are doing everything correctly!).</p>
<p>So, how would backtracking actually look on this question…</p>
<p>(C) a = 3/2 = 1.5 (if you convert to decimals, it’ll be easier to use your calculator!)
[1.5] = 1.5^2 - 1.5 = 2.25 - 1.5 = .75
[1.5 - 2] = [-.5] = (-.5)^2 - (-.5) = .25 + .5 = .75</p>
<p>(Note that jamesford’s note to attack it one piece at a time comes in very handy in this solution, as well!)</p>
<p>And there you have it…when a is 1.5 or 3/2, [a] is equal to [a - 2]!</p>
<p>Final Note:
The backtracking technique, especially on this particular question, may take a bit of time and, obviously, would have required even more time if (C) had not been correct, so the algebraic solution that jamesford provided is probably more efficient and perhaps easier for many students. </p>
<p>However, keep in mind two things… </p>
<p>1) with more practice with the technique, you will become more confident and efficient in using it and…</p>
<p>2) in circumstances where you find the algebraic solution difficult to “see” or if you try a question algebraically but you get stuck, you are going to want to know this alternate solution so you are not forced to sacrifice any points unnecessarily.</p>
<p>awesome, thanks for the help!</p>
<p>Let be defined as = x^2 - x
for all values of x. if [a] = [a-2]
what is the value of a?</p>
<p>Since = x^2 - x</p>
<p>Then, [a] = a^2 - a</p>
<p>And [a - 2] = (a-2)^2 - (a-2)</p>
<p>Thus,</p>
<p>a^2 - a = (a-2)^2 - (a-2)
a^2 - a = (a-2)(a-2) - a + 2
a^2 = (a-2)(a-2) + 2
a^2 = a^2 - 2a - 2a + 4 + 2
a^2 = a^2 - 2a - 2a + 4 + 2
a^2 = a^2 - 4a + 6
0 = -4a + 6
4a = 6
a = 6/4
a = 1.5</p>
<p>Which is C</p>