In Exercises 35–46, determine the constant that should be added t...

Question

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.

x^2 - (2/3)x

Accepted Answer

Welcome back. I'm so glad you're here. We have this expression, we have X squared minus 8/ X. And we are to complete the square. Well what we've got here is we have an A X squared plus B. X. And are A corresponds to one in front of the X squared and B corresponds to that negative 8/7. Well this is great because when we're completing the square, the first thing we want to do is ask ourselves does a equal one and yes in this case it does. And that makes for a lot fewer steps. And the next thing we're going to ask ourselves is well what is that? B term that be term is negative 8/7. We're going to take the B term. We're going to divide it by two and then we're going to square it. So the b term negative 8/ divided by two squared. Let's do the simplification. I'll rewrite this negative 8/7 divided by two like this because that helps me remember. This is the same as negative 8/7 times one half. And doing that, multiplication will get negative 8 months and we can simplify that one more time. We'll have a negative four seventh that needs to be squared. And now I want to take note of this number that we have inside the parentheses once it's simplified before we've gone and squared it Because um that number will be used again in a moment but then once we go through and square it we are going to get a positive 16:49ths and then we're going to take this simplified number and we are going to both added and subtracted to our expression. So we'll have an X squared minus R. 8/7 X. And first will add R. 1649th. And then we're also going to subtract it because 1649 -1649. Well that's nothing. So in effect we're doing nothing. Right. Well what we're going to do next is we're going to group these 1st 3 terms and we are going to factor them. So we look at them and we say well what times what gives us X squared? That's going to be an X. Times and X. And what times what um what two factors when multiplied will equal 1649th. But when added will equal -8/7. Well, that's the work that we just did in green to complete the square. And that is that number that we highlighted. That is the factor that when multiplied gives us a positive 16 49th and when added to itself gives us a negative 8/7. So we'll put minus 4/7 minus 4/7 and x minus 4/7 times X minus 4/7. That is the same as x minus four for seventh squared. Well that part's done but we absolutely cannot forget to bring down this number that we kind of shunned and stuck off to this side. This negative 16 49th and minus 16 49th because this is all part of our completing the square, we look at our answer choices and that matches with answer choice, a well done, we'll catch you on the next one.

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