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Completing the Square - Problem 3
A lot of times when you're doing math problems they tell you what process to use, but sometimes they don’t. Let me look at this one and I’m going to tell you about a big hint that I see.
First thing it tells me to solve this problem by completing the square. Okay I know what process I’m supposed to use. If they didn’t tell me that however, I could keep reading and it says round your answer to the nearest 100. That tells me I’m probably going to have some kind of nasty decimal solution which tells me I wouldn’t want to graph because graphing isn't very precise and I also probably wouldn’t want to try factoring because factoring only works effectively if we have whole numbers or fraction.
So this tells me I’m going to want to probably complete this square or use the quadratic equation just because it says round to the nearest 100. Okay, so it told me to complete the square. The way completing the square works is first thing you need to do is make sure your leading coefficient is 1. It is, so that’s good. The next thing I want to do is rewrite it so that my x terms are together and that my constant term is on the other side of the equals.
I personally like to write little blanks in here because I’m going to be adding something to both sides of the equation and here’s how I'll find it. The way to complete this square is to take half of your b term, so half of -10 is -5, then since I’m squaring it, -5 times -5 is +25. I can’t just add 25 there I have to add 25 to both sides of the equation in order to maintain that equality. So this side is now a perfect square trinomial, there it is and it’s equal to 65. That’s the completing the square step.
From here I’m just going to solve for x by taking square roots. Take the square root of both sides so I’ll have x minus 5 is equal to the positive or negative square root if 65, and then what I’m going to use is a decimal approximation they told me to use the nearest hundredth so on my calculator when I type in square root of 65 I get 8.06. So I'll have 8 minus 5 equals 8.06 and also x minus 5 equals -8.06.
When I add 5 to both sides here I’ll get x equals 13.06 that’s one of my solutions approximated to a hundredths, and then here when I add 5 to both sides I get x equals -3.06. Those are my two decimal approximations to check them I could plug them in as my x numbers, take away 40 and I won’t get exactly 0 because I did some rounding. I’ll get something really close to 0 that’s what an approximation is.
So again completing the square is useful anytime your directions hint that you're going to have a decimal answer, completing the square is good or the quadratic equation would be good in that situation, but in this case I use completing the square because they told me to.