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The Discriminant - Problem 2
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When you’re doing all these problems about quadratic equations it can be really hard to remember that math really does show up in real life. A lot of times your teachers will assign word problems but I know you guys, you’re probably going to skip them. What I want to do is show you that these quadratic equation problems really do apply to real life, for example in business.

A furniture business uses the formula f(x)=5400+300x-40x² to calculate its monthly income. Where f(x) stands for income and x represents the number of items sold. Will their monthly income ever reach $6000? Okay you can solve this problem using the discriminant and here’s how.

What I’m going to do is set up and equation using both of these pieces. They told me income is f(x), right? F(x) stands for income. So right there where it says f(x) I’m going to write my income number, 6000. I’m going to have that equal to 5400 plus 300x minus 40x². I’m going to solve, put this in standard form equal to zero, find the discriminant and see what that tells me. If I subtract 6000 from both sides, then it will look like this, -600 plus 300x minus 40x². I’m going to rewrite this so that it's in order of decreasing exponents, so I can find my a value, my b value and my c value really easily.

Okay so now I can see a equals -40, b equals 300 and c equals -600. Let’s plug it into out discriminant formula and see what we come up with; b² minus 4ac. So I’ll have 300² take away 4 times -40 times -600. Stick that in your calculator. Give me a second while I do that. 300² minus 4 times -40, careful with parenthesis, times -600, okay and I got -6000. I have a negative discriminant. What this tells me is “no”. Negative discriminant means no solutions. Never, ever, ever will this company reach their goal of $6000 a month. It doesn’t have to do with the fact that I got 6000, here; the important part is I got a negative value for my discriminant. That tells me no, there’s no solutions.

If I were to graph this, and you don’t have to graph this but if I wanted to graph this if you’re a visual learner, if you graphed their function it looks something like this, 6000 is like just barely above. I don’t know if you can tell on your small screen but I drew this so that these lines don’t actually cross. This company makes really close to 6000 but they don’t actually hit it. And I was able to find that out using the discriminant without having to do any graphing. It’s pretty cool once you get the hang of the discriminant and how it applies to real life. Some day you guys might own a business and you might be able to come up with an equation and then predict what your monthly income might be based on that.

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