Video Transcript
The given figure shows the graph of ๐ of ๐ฅ is equal to ๐ฅ cubed minus four ๐ฅ squared plus one. Use the graph to determine the number of solutions to the equation ๐ฅ cubed equals four ๐ฅ squared minus one. Use the graph to determine the intervals in which the solutions to ๐ฅ cubed equals four ๐ฅ squared minus one lie.
To answer this question, weโll begin by considering the equation ๐ฅ cubed equals four ๐ฅ squared minus one. Weโre looking to find the number of solutions to this equation. And if we were going to solve the equation, we would solve it a little like solving a quadratic. Weโd get an expression in terms of ๐ฅ that is then equal to zero.
To achieve this, weโll begin by subtracting four ๐ฅ squared from both sides of our equation. That gives us ๐ฅ cubed minus four ๐ฅ squared equals negative one. Weโll then add one to both sides. And weโve now achieved an expression in terms of ๐ฅ that is then equal to zero. So we have ๐ฅ cubed minus four ๐ฅ squared plus one. And thatโs equal to zero.
So whatโs the relationship between this equation and the equation for the graph given. Well, in the original equation for the graph, we have ๐ of ๐ฅ equals ๐ฅ cubed minus four ๐ฅ squared plus one. But weโve made ๐ฅ cubed minus four ๐ฅ squared plus one be equal to zero. So ๐ of ๐ฅ has been replaced with zero. ๐ of ๐ฅ is the output of the graph. And we often call it ๐ฆ. So where does ๐ฆ equal zero on our graph?
The equation ๐ฆ equals zero is also known as the ๐ฅ-axis. So to find the solutions for ๐ฅ cubed equals four ๐ฅ squared minus one, we need to find the points on the graph where ๐ฆ or ๐ of ๐ฅ equals zero. And thatโs the points where the curve intersects the ๐ฅ-axis. We can see that this happens on one, two, and three separate occasions. So this means the equation ๐ of ๐ฅ equals zero or ๐ฅ cubed equals four ๐ฅ squared minus one has three solutions.
The second part of this question asks us to determine the intervals in which the solutions lie. And we can make estimates to the solutions by reading the points where that graph intersects the ๐ฅ-axis. So letโs consider the scale on the ๐ฅ-axis. We see that five small squares represent five units. And if we divide through by five, we see that one square represents one unit. So we can add the values negative two, negative one, one, two, three, and four onto our graph as shown. And if we look carefully, we can see that our first solution lies roughly halfway between negative one and zero. So we say that ๐ฅ is greater than negative one and less than zero.
Itโs important here that we use a strict inequality because we can see that it definitely canโt be equal to negative one or zero. The next solution lies roughly halfway between zero and one. So we say that that solution is greater than zero and less than one. And our final solution lies between three and four. Itโs closer to four than it is to three. But it doesnโt quite hit four. So we say that this solution ๐ฅ is greater than three and less than four. So the intervals in which the solutions lie are ๐ฅ is greater than negative one and less than zero, ๐ฅ is greater than zero and less than one, and ๐ฅ is greater than three and less than four.
Now, there is a way we can double check these intervals. Letโs take, for example, ๐ฅ is greater than three and less than four. We can substitute ๐ฅ equals three and ๐ฅ equals four into the original function. And if we substitute three in, we get negative eight. And if we substitute one in, we get four.
Remember we said the solutions lie at the point where ๐ฅ cubed minus four ๐ฅ squared plus one is equal to zero. This sign change indicates to us that there must be a solution within this interval. And the graph is continuous, which is important because it means it doesnโt stop. We can guarantee that there will be a solution in the interval ๐ฅ is greater than three and less than four. And we can repeat this process if we needed for ๐ฅ is greater than negative one and less than zero and ๐ฅ is greater than zero and less than one.