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The Quadratic Formula - Problem 2
Depending on what time of year you learn this process in your Math class, you may or may not be asked to find exact solutions in simple radical form. I’m going to do this problem two ways. First I’m going to do it and give you the decimal answer which is kind of like the more basic way I guess, and the other way I’m going to show you is how to find the exact solution using simplifying square roots. You’re going to have to ask your teacher which method he or she wants you to use. It depends on what you’ve already learned in your class.
Okay so here we go. I have an equation I want to go through and use the quadratic formula. So here it is, you guys had it memorized, I’m not going to sing it this time because you’re lucky. -b plus or minus square root of b² minus 4ac all over 2a, okay? What that means is each of those letters stands for a coefficient so I’m going to start with negative in front of my b term, -(-2) is +2 plus or minus means I’m going to have two different answers b² will be 4 take away 4 times a which is 7 times -8 all divided by 2a. Please make sure the process gets divided by 2a.
Okay next thing I’m going to do is simplify under that square root there. X is going to be 2 plus or minus. I got to simplify whatever 4 times 7 times -8 is. That’s 244. So what I really have is 4 minus -244 which is 4 plus 244 which is 248. All divided by, my a term was 7 so on the bottom of my fraction I’m going to have 14.
Okay, what I’m doing from this step on is all simplifying. This is where I’m going to split so either pay attention now this is how you do it using decimals to get an approximate answer, or else tune out and plug your ears for 30 seconds until we do it using simple radical form to find the exact answer depending on what your Math teacher wants.
Okay so if I’m doing this using decimal approximation, I’ll pull out my calculator and find out that the square root of 248 is 15.1. So I’ll have 2 plus 15.1 then divided by 14 and also I need to do 2 minus 15.1 then divide by 14. Be careful with order of operations, simplify the tops before you divide by 14. This first problem I’m going to get the answer x equals 1.22, second one I’m going to get x equals -.94.
Okay, that’s how you get him decimal approximations. Here’s another way. If you know how to simplify square roots, you probably know how to do the square root of 248 as equal to the square root of 4 times the square root of 62. I broke that down into its square roots. So if I were to write the square root of 248 in simple radical form, it would be 2 root 62. So my problem now looks like 2 plus or minus 2 root 62 divided by 14. This is how you find the exact solution.
I’m going to simplify the top by factoring out 2, 1 plus or minus root 62 divide by 14. I can reduce that 2/14, so here’s what my final solution is going to look like in simple radical form and this is the exact answer.
The cool thing about this red answer is that it’s exact. This blue guy I had to cut off some big long decimals this is exact. Also notice here I wrote them out as different answers here I kind of wrote it as one answer with that plus or minus business that’s okay. So please ask your teacher before you do this homework if he or she wants you to do it in simple radical form or with the calculator and decimal approximations. Some teachers make you do both. That’s what I do.