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 - 7x

Accepted Answer

welcome back. I'm so glad you're here. We have this expression Z squared minus five Z. And I will try to write carefully but my twos and my Z's might kind of blend together so I'll do my best So we are to complete the square with this expression. When we're doing that, we look at what we've got, we've got something in the form of A X squared plus B X. Well our X's or Z's in this case and A corresponds to the number in front of the second degree term and B corresponds to the number in front of the first degree term. And the first thing we ask ourselves is does a equal one and in this case, yes it does. That makes it for far fewer steps with what we're doing here. So because A equals one, the next thing we're going to do is we're going to take our B term, We're going to divide it by two and we are going to square it. So our B term again is negative five. We're going to take negative five divided by two and square it. And once you've got something simplified in the parentheses there and before you square it, I want you to take note of what that simplified number is and then move on because we do need to square it. So negative five halves squared is going to be a positive 25 4th. So once we've gone through this work, we're going to take that number and we're going to both add it and subtract it to our expressions. So we'll have Z squared minus five Z plus 25 4th minus 25 4th. And because we have a plus 25 force -25 horse and that equals zero. We in effect haven't changed our expression at all. Now we're going to take these first three terms and group them together and we are going to factor them. So we ask ourselves and be sure you bring down that -25/4. We're going to ask ourselves what times what gives us Z square. That's going to be Z times Z. And what? Two factors when multiplied? Give us a plus 25 forts and when added give us a negative five. Well that's the work that we just did before. That's that highlighted number that we took note of. That's going to be a minus five halves and a minus five halves. And we can check this real quick with foiling it out to make sure Z times Z is Z squared our outsides. Z times negative five halves is negative five halves. Z Oh my Z's in twos this is a disaster. Are negative five halves and z's are gonna be again negative five halves. Z. And our last negative five halves times negative five halves is going to be plus 25 4th combining our like terms in the middle will have a Z squared minus 10/2 Z plus 25 4th and simplifying this fraction again is Z squared minus five Z plus 25 4th and as I said and then didn't do we still have to bring down that negative 25 fourths And look this does match the expression that we had up here. So we factored correctly because z minus five halves times, Z minus five halves is the same as z minus five halves squared. We can rewrite it as such and then just don't forget our minus 25 4th and that is our expression. Having completed this square and done our perfect square try no meal and factor we look at our answer choices and that matches with answer choice C. Well done, we'll catch you on the next one.

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