One of the common challenges with quadratics is in solving word problems. Fortunately, most of them sort into one of two categories. Ones where you are looking for the vertex, and ones where you are looking for the zeros. The following examples and practice are some typical quadratic word problems although there are many, many examples of problems that require quadratics to solve.
This profit optimization problem has us look at a situation where as you rise the price you lose customers, but as you lower the price you gain customers. At the extremes you can predict that you will raise the price so high that you would lose all customers, or you would lower your price until it is free. Either way you make no money. Somewhere in the middle is the perfect price.
Questions like this can always have added complexities, but for the purposes of this practice we will assume that you earn 100% profit from your sales. That means that our Profit = (# of sales)x(cost per item) As we introduce a variable into each of those terms, we create our quadratic. If you would like to see an example of one such problem worked out, check out the video below.
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Ready to try out some on your own? You might want to grab yourself a piece of paper before you get to work on these.
The next type of problem is one that makes use of Pythagorean theorem. We will be setting up a right angle triangle however as we use a² + b²= c² we will introduce a quadratic problem in which we need to solve for the zeros rather than the vertex. Check out the video below to see this type of problem.
Time to try some on your own. Keep in mind what the variable x represents in this problem. It will represent hours. Once you get your final solution you will need to use that number in order to determine the time of day to solve the problem.
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The following two practice problems involve throwing a ball. Kinematics is a very common theme of quadratics because gravity causes objects to fall in a parabolic shape. Take note that in the two different problems, one is interested in a maximum height, (vertex) and the other is looking for when the ball lands, (zero).
The first practice is looking for the maximum height of the ball. The hint in these questions is that it is asking for a MAXIMUM.
The second ball launch practice is looking for when the ball lands. The hint in these questions is that it is asking for when the height is ZERO.
If you’ve managed to take care of the practice above, another common quadratic problem is the sum and product. Try to use what you have learned from the problems above to solve this using the substitution technique and quadratics, rather than guess and check.
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This final practice has us looking for the length of a fence in order to build a fence of a specific size. There has been no mention of solving for the maximum area, so that is a good hint that this type of problem is not searching for a vertex as a solution.
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