Two Deadly Algebra Mistakes That You Need To Fix Right Now Today I would like to go over two very common mistakes that students make when solving algebra problems that lead them to the wrong answer. These two errors are very basic and easily fixed. But surprisingly, even students in college calculus classes are still making these mistakes. Many students will make one of these mistakes on their SAT, ACT, or GRE, thus lowering their score. The good news is that if you just take a few minutes to read this article carefully, you will probably never make these two mistakes again. So please, don’t be another statistic. Read and absorb the information in this article right away. And if you want to discuss other common algebraic mistakes, please bring them up in the comments below. Algebra Mistake Number 1 Let me start with a very simple math problem. What is –32 equal to? Please take your time to answer this question yourself before looking at the answer. Is the answer 9 or –9? The answer is in fact –9. Most of the time when I tell students the answer they say “but I thought when you square a negative number it becomes positive.” And in fact, they are right. When you square a negative number it does become positive. But that is not what is going on here. Don’t worry if you got this wrong. It does not mean you are not intelligent. It has nothing to do with your problem solving skills. It has only to do with a very small gap in knowledge which we will correct right now. There are actually two issues that need to be addressed here. Point 1: The first thing that you need to understand is that the expression –x is equivalent to the expression (–1)x. In other words, the operation of negation is equivalent to multiplication by –1. Point 2: The second thing that you need to understand is the basic order of operations. In particular, exponentiation is always done before multiplication. Note that the order of operations that we as humans currently use is a completely man-made construction. There is no really good reason that we do one operation before another one, except that a long time ago someone decided that this is the way we are going to do things. So now, let’s get back to our computation. By Point 1 above we have –32 = (–1)32. By Point 2, we must perform the exponentiation first. So we have –32 = (–1)32 =(–1)9 = –9. Note: Although –32 = –9, we do have that (–3)2 = 9. The reason is that in the latter case we are squaring a negative real number, and squaring a negative real number does always give a positive number. Let’s try another example involving functions: If f(x) = –x2 + 5, then what is f(2)? Solution: f(2) = –22 + 5 = –4 + 5 = 1. Did you get that one correct? If not, reread this section until you can explain your error. Try this similar question: If f(x) = –x2 + 5, then what is f(–2)? Solution: f(–2) = –(–2)2 + 5 = –4 + 5 = 1. Note that the solution to this question is pretty much identical to the last one. This is because (–2)2 is the same as 22. See the note above if you are still having trouble with this. Algebra Mistake Number 2 Let’s look at another pretty basic math problem. Can you expand the expression –(x + 2)? Again, please take your time to answer this question yourself before looking at the answer. Did you get –x + 2, or –x – 2 ? The answer is in fact –x – 2. When negating an expression with two terms, many students forget to distribute the minus sign correctly. In this case, both terms need to be negated. If it helps you, you can rewrite –(x + 2) as (–1)(x + 2). Then we get (–1)x + (–1)2 = –x – 2. If you are okay with distributing in general, but only have trouble with this specific instance, then this might help. If this doesn’t help, then you need to go over the distributive property very carefully. As another example, try to simplify this expression: (x – 3) – (x – 5) Let’s begin by removing the parentheses and distributing the subtraction symbol in the middle correctly: x – 3 – x + 5 Finally we combine like terms to get an answer of 2. Avoid Making Other Common Mistakes On Standardized Tests If you want to learn how to avoid all the traps that occur in the math sections on standardized tests such as the SAT, ACT, and GRE, as well as learn all the most important math strategies, then take a look at the Get 800 collection of test prep books. Click on the picture below for more information about these books. If you have any questions on this topic, then please post them in the comments below. Speak to you soon! Comments comments