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Applications of Quadratic Equations - Problem 1
You guys are getting to be really good Math students so you already know a lot of skills about how to approach a word problem. It’s really important that you read carefully and if you can try to underline or at least remember in your head what looks like important information. Here’s an example; suppose a rectangle has an area of 90m² and the sides are x plus 1 and x plus 2, find the dimensions of the rectangle.
Okay so in my head I’m already thinking about some important words like area of a rectangle. You guys already know how to find the area of a rectangle; it’s length times width. So I’m going to draw a little picture, it’s always a good strategy and instead of length and width, I’m going to label those sides with what they told me: x plus 1 and x plus 2. And then I’m also thinking about area of a rectangle means multiply the length times the width. That’s how I’m going to set up my problem. Area means multiply length times width. So to do this I’m going to multiply x plus 1 times x plus 2 that’s going to be equal to my given area of 90. For many students that’s the most difficult part. If you can set it up like this, it’s just a standard solving quadratic. You have a choice of, after you get this all distributed by using FOIL:, you have a choice of graphing, quadratic equation, factoring, completing the square, taking square roots, blah, blah, blah. You have lots of different options for how to approach this problem depending on what your own personal strengths are.
The first thing I’m going to do is FOIL and then set this equation equal to zero so I can begin the solving process. FOIL, my firsts, outers, inners and then lasts combine like terms here on these middle guys. So I’ll have x² plus 3x plus 2 and at this point it’s still equal to 90. So that is kind of tricky because for most of my solving techniques, I want his trinomial to be equal to zero. So the next thing I’m going to do is subtract 90 from both sides so that I’ll have x² plus 3x take away 88 equals 0. Okay now I’m ready to solve.
Quadratic formula, completing the square, blah, blah, blah, factoring is my own personal favorite. I’m trying to think of 2 numbers that multiply to negative 88 and add up to 3. Factoring was a good method in head in this situation because my leading coefficient here is 1. Sometimes factoring can be easier if you have a leading coefficient of 1. If it’s not 1, factoring can be more difficult. So let’s see numbers that multiply to negative 88 and add up to +3, I’m thinking its going to be 11 and -8 because 11 times -8 is negative 88 and then if I add those guys together I get +3.
Okay, next thing I want to use is the zero product property to help me find x and I’ll see my x values are -11 and x equals 8. A lot of students will stop here, they’ll turn in their homework, and they’ll say I’m done. I’m an A+ student and they’ll think they did this all right and that’s kind of right, everything we’ve done so far is correct but we haven’t totally answered the question.
The questions said find the dimensions of the rectangle. Dimension means the side length. So what I need to do is go back to my original picture where I labeled my side length x plus 2 and x plus 1 and think about what my answers could be; what my dimensions could be. First thing, I have an answer that is -11 so I could have -11 plus 2 but that’s -9 and -10. It doesn’t make sense to me to have negative lengths so that tells me this is not going to be one of the answers to my problem. It’s not going to be one of my solutions. It doesn’t make sense in the real world to have a negative side length. So that’s not going to help me find the answer. My answer for my dimensions is going to come form this. My side lengths are going to be 8 plus 2 and 8 plus 1. Those are my two side lengths. I stuck ‘m’ out there to represent this problem was in meters and before I move on, before I turn in my homework I want to make sure I do this correctly by checking.
So let’s see, let’s just try to check. If I had 10 and 9, let’s just make sure that’s the same thing as some number plus one and some number plus two. I know that’s right because it's 8 plus 1 and 8 plus 2 and the area of the rectangle would be 90, sweet. That means I did it correctly.
Please you guys the last thing I want you to remember from this video is how important it is for you to answer the question. A lot of times in multiple-choice test teachers are tricky and they’ll put the answer as -11 or 8 in the possible answer choices, whereas really if you had answered this question the correct answer would be both 9 and 10.