Use technology to plot the graph of 𝑓 of 𝑥 equal to 𝑥 cubed plus five 𝑥 squared minus 100, and use this graph to find the solutions to the equation 𝑥 cubed plus five 𝑥 squared equals 100 to two decimal places.
Before we do any graphing, I want to identify both of these equations. Our first equation is 𝑓 of 𝑥 equals 𝑥 cubed plus five 𝑥 squared minus 100. When we go to identify our second equation, we need to be a little bit more careful.
The function of 𝑥 the 𝑓 of 𝑥 for our second equation is equal to 𝑥 cubed plus five 𝑥 squared. What kind of relationship did these two functions have with one another? This means at every point our pink line will be 100 units below our yellow line. Our yellow line crosses the 𝑥- and 𝑦-axis at zero, zero. And our pink line has that 𝑦-intercept at zero, negative one.
If we’re looking for the solution of our yellow line at the point 100, we want to know when is it at a height of 100. We can look at the pink line at point zero because the pink line is 100 units below the yellow line. When the pink line is equal to zero, the yellow line would be equal to 100.
Now is where the technology will come in. I just use an app on my iPad and then I graphed this pink line 𝑥 cubed plus five 𝑥 squared minus 10. It gave me this image. Based on what we’ve already said, the point at which this line crosses the 𝑥-axis will be the solution to 100 equals 𝑥 cubed plus five 𝑥 squared.
What we should do now is use our technology to zoom in on this intersection. Once we zoom in, we see something that looks like this: the intersection is between three and three and a half. It’s very close to three and a half. We can zoom in a third time. And then whether you are using a calculator or an app or a website, you can ask it to tell you what that intersection is.
The intersection of the pink function and the 𝑥-axis is at 3.441779 continuing. And at this point, where our pink function crosses the 𝑥-axis will be the point where our yellow function crosses 100. The solution for the pink function equal to zero is the solution for the yellow function equal to 100. So we take this value 3.441 and round it to two decimal places. The place where 𝑥 cubed plus five 𝑥 squared is equal to 100 is about 3.44.
If you weren’t completely sure about this process, something that you can do afterwards to confirm that you found the right answer is to take 3.44 and plug it in to that equation. Yes, 3.44 cubed plus five times 3.44 squared equal to almost 100. When you do this, you’ll find that the statement is true. And we’ve found the solution 3.44.