在本课中，我们将讨论一些涉及代数的定量比较问题的策略。 First of all, many of the important algebra patterns discussed in the Algebra module remain important here. 所以我们假设您熟悉代数模块中的所有视频。在哪里，显然，在这里没有完整的代数审查， 因为它都覆盖在那里。

We're just talking about how these things play out in the QC questions. So, the three big patterns to know. The square of a sum. Remembering that r plus s quantity squared, that's not just r squared plus s squared. We have to FOIL out r plus s times r plus s.

它等于这种模式。非常相似的模式，平方差异。 然后可能是最重要的，两个方块的差异，r平方数减去s平方，以及如何考虑。 所有这三个都是非常重要的因素，可以很好地了解，而这些是他们喜欢在定量比较中进行测试的因素。

For example, here's a practice problem. Pause the problem and then we'll talk about this. 好的，请记住，只要我们可以通过同一件事乘以数量 知道这是积极的。嗯，a，b和c，这些都是长度。

显然，每个长度都是正数。我们添加了两个长度，一个加c。 这必须是积极的。所以我们将乘以该分母，即加c。 所以在左边，我们只是得到B个平方。在右边，我们得到了两个方格模式的差异， C减去次数c加a。

And of course, that's gonna equal c squared minus a squared. Well, then just to get everything positive, we'll add a squared to both columns. And then we get a squared plus b squared on the left, and c squared on the right. Now remember, a squared, a and b, these are the legs of our right triangle. c is the hypotenuse.

And of course, Mr. Pythagoras guarantees that for any right triangle, a squared plus b squared would have to equal c squared. That is the Pythagorean theorem. So, these two columns are always equal as long as the angle's a right angle. And of course, the diagram does guarantee that it's a right angle. So always equal, the answer is c.

这是另一个问题。 Pause the video and then we'll talk about this. 现在，您可能会诱惑在此处插入数字。并堵塞数字实际上可能导致一些问题。

相反，我要说的是，如果你有代数表达式，请尝试简化它，就像你解决方程一样。 It's very, it's easiest if you can get x in one, one of the two quantities. Just have that equal x by itself. 所以你比较X到一个数字。所以我要做的第一件事就是我会从两种数量中减去x。

And of course, whether x is positive or negative, I'm allowed to add or subtract it. 这完全没问题。所以我减去了它。 那么我要做的就是从两侧减去2。然后我会划分6。

So that gives me x in Quantity A and 13 over 6 in Quantity B. Those are the two things we're comparing. So if x is greater than 2, which of these two quantities is bigger? 好吧，注意一些事情。请注意，例如，x可以等于13 6因为13岁以上自身是一个略大于2的数字。

Again, there's no guarantee that x is an integer. So don't add that restriction here. X可以是任何数字，所以x可以是13超过6.在这种情况下，两列是相等的。 If we picked 3, for example, and plug in, then Quantity A is 23, Quantity B is 18, so A is bigger.

所以，两个不同的数字给我们两个不同的答案。事实上，如果我们选择了不到6岁以上的数字，但 larger than 2, say 2 and one-hundredths, something like that, then it would actually turn out that quantity B is bigger. 因此，我们可以获得我们想要的任何关系，大于，小于或平等，显然没有持续的关系。

答案必须是D. Here's another practice problem. Pause the video and then we'll talk about this. Okay, here also, it would be kind of a mistake to plug in numbers.

In other words, if you plug in numbers, you'll always find that Quantity A is bigger, but plugging in a bunch of different numbers is not gonna help you too much. Instead, notice that that expression in Quantity A is very close 对于我们的代数模式之一，总和，总和的平方。特别是，如果我们有x平方加16x加64， that would be a perfect square.

So let's just rearrange it a bit, okay? So we'll separate that 67 into 64 plus 3. And then that x squared plus 16x plus 64, that is the sum, that is the square of a sum. 所以这是x加8平方。这就是为什么认识这些模式非常重要。

That's a perfect square. So in other words, what we have here is x plus 8 squared plus 3. 现在，把它放在那种形式，让一些东西很清楚。当某事被平方时，它总是积极的，或者它可能是零。 If x equaled negative 8, that would equal zero. When we add 3, it's always greater than 0 because even when x equals negative 8, we'd get 0 squared, which is 0.

But then plus 3, that would still be 3. That would be greater than zero. 所以它是3加0或3加上一个正数。这总是大于0，所以回答A实际上总是正确。 Here's a practice problem.

Pause the video and then we'll talk about this. Okay, this is one that involves function notation. So don't be intimidated. Sometimes they will throw function notation at you. They define the function notation.

And c is a number such that f of c is 0. So let's think about this. C的F等于0.换句话说，这意味着C平方减去25等于0。 Well, if we rearrange a little bit, that would mean that c squared equals 25, and we have to be very careful here.

c等于什么呢?是的,可能是5,但是to keep in mind when we have, 采用变量的平方根，我们必须包括加号或减号。所以C可以是5或负5。 那些是两个数字，当平方时，相等的正面25。所以如果C是加5或减5，那么我们就无法确定是否 Quantity A is bigger or smaller because if it's 5, it's bigger than 3.

如果它是负5，它小于3.所以，它可以到哪一种方式。 请注意，如果您忘记采取，忘了包含加号或减号，您就会让错误的答案。 This question is, in disguise is actually testing whether you remember that plus or minus sign.

因为有一个加号或减号，我们无法决定关系，所以答案是D. Here's another practice problem. Pause the video and then we'll talk about this. Sometimes, when you're given one of these very complicated equations before 比较，它有助于使用该等式工作并简化它。

So the first thing I'm gonna do is I'm just gonna subtract w to get the fraction all by itself. 然后请注意，我在分母中有一个m plus w，另一边是m pigus w，所以这很暗示。 I notice if I multiplied those, I'd have the difference of two squared patterns. So multiply by that denominator.

所以，我只是在那边得到k个平方加1，然后我得到两个方形图案的差异，m减去w倍。 And of course, that's going to be m squared minus w squared. Then just to get the, the k and the w on the same side, I'm gonna add w squared to both sides. So now I have k squared plus w squared plus 1 equals m squared.

So this looks an awful lot like the, the two quantities we have here. And I'll just point out, with any number on the number line, 如果我们添加1，那么无论是积极还是负面，我们都会更大。所以对于任何一个，例如，称之为一个数字，P Plus 1总是大于p。 无论是积极还是负面，无论是整数还是分数都无关紧要，这无关紧要，这是真的 every single number on the number line.

所以不管是什么k平方加w，实际上我们知道它必须是某种积极的数量，如果我们加1，那就更大了。 当我们添加1时，即等于m，所以这绝对意味着m平方是比k平方加上的1大。 因此，该数量B总是更大。 So be familiar with these important patterns, the Square of a Sum, the Square of a Difference, and the Difference of Two Squares.

The folks who write the test really love these patterns. It may help to treat the two quantities as the two sides of an equation, and 孤立x。这通常是一个很好的伎俩， 特别是如果您在两种数量中具有相对简单的线性表达式。记住当您拍摄变量平方的平方根时，请记住加号或减号。

And always remember the basic rules of inequalities. As we've said above, understanding inequalities is essential for 了解GRE定量比较的深度逻辑。