### Video Transcript

Find the solution set of two π₯ cubed equals 32π₯ in the real numbers.

First, we should get our equation equal to zero. And we want to keep the variable with the highest exponent positive, so π₯ cubed β that needs to stay positive. So the two π₯ cubed will stay positive. So we need to subtract the 32π₯ from both sides of the equation, which leaves us with two π₯ cubed minus 32π₯ equals zero.

So since there are two terms, we can factor by first taking out a greatest common factor. And we can take out a two and an π₯. If we take out two π₯ from two π₯ cubed, weβre left with π₯ squared. If we take out two π₯ from 32π₯, weβre left with 16.

Now, inside the parentheses, we have a difference of two squares because π₯ squared we can square root and 16 we can square root. The formula for difference of squares is π squared minus π squared equals π plus π times π minus π. So if we have π₯ squared minus 16, the square root of π₯ squared is π₯ and the square root of 16 is four. So we will have π₯ plus four and π₯ minus four.

So now, weβve completely factored our equation. So to find the solution, we need to take each factor and set it equal to zero. So we set two π₯ equal to zero, we set π₯ plus four equal to zero, and we set π₯ minus four equal to zero.

So for two π₯ equals zero, we need to divide both sides of the equation by two. And we get that π₯ is equal to zero. For π₯ plus four equals zero, we need to subtract four from both sides of the equation. And weβre left with π₯ equals negative four. For π₯ minus four equals zero, we need to add four to both sides of the equation. And we have π₯ equals four.

So our solution set would be zero, four, and negative four. And the order does not matter.