Taking A Guess – A More Difficult Example Yesterday I described a very simple yet effective strategy to solve certain math problems on standardized tests such as the ACT and SAT. The strategy of “taking a guess” is a very useful way to avoid complicated algebra and ensure that a question is answered correctly. See the post from last week for details if you have not already been exposed to this strategy: Basic SAT Math Strategy – Take A Guess Example Let’s take a look at another example of how this strategy can be used effectively. Here is a more difficult problem for you to chew the end of your pencil over: Let the function f be defined for all values of x by f(x) = x(x + 1). If k is a positive number and f(k + 5) = 72, what is the value of k? Looks tricky? Well let’s see how we can solve this tricky problem quickly by “taking a guess.” Let’s take a guess for k, say k = 2. Then we have f(2 + 5) = f(7) = (7)(8) = 56. This is too small. So let’s guess that k = 3 next. Then f(3 + 5) = f(8) = (8)(9) = 72 which is correct. So the answer is 3. Algebraic Solution You can see that by using the method of guessing we have found the answer very quickly. Let’s compare this to solving the problem algebraically. Do not worry if the solution that follows confuses you. You will never have to use this method if you choose not to. f(k + 5) = (k + 5)(k + 6) = k² + 11k + 30. Since f(k + 5) = 72, we have k² + 11k + 30 = 72. Subtracting 72 from each side of this equation yields k² + 11k – 42 = 0 (k – 3)(k + 14) = 0 So k = 3 or k = -14. We reject the negative solution because the question says that k is positive. Therefore the answer is 3. Further Discussion As you can see, the algebraic solution involves a fairly complicated multiplication which leads to a quadratic equation. The simplest way to solve this quadratic equation is by bringing everything over to one side and factoring. We can see that the answer is obtained correctly, but for most students there is a danger of getting lost in the algebra. This method also takes longer than the method of taking a guess. So for these reasons, why would you want to use algebra to solve a problem such as the example above? Knowing how the algebra works, however, is good for gaining mathematical maturity. This is certainly important if you want to get an 800 in SAT math or a 36 in ACT math. So if you are going for a perfect score, practice the algebraic solution at home. Just do not use it on test day! More Practice If you have any questions or comments regarding this strategy, please do let me know. And if you want more practice with this particular strategy, as well as the other important strategies you need to know to improve your standardized test math score, please check out the Get 800 collection of test prep books. Click on the picture below for more information about these books. Make sure to join the newsletter above to receive more of my strategies, hints and tips as you prepare for your standardized test! Comments comments