In Exercises 23–34, determine the constant that should be added t...

Question

In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.
32. x²+4/5x

Accepted Answer

Welcome back. I am so glad you're here. We are given this problem and we're told by completing the square, identify the constant needed for the given expression to become a perfect square trinomial also write the trinomial and express it as a square of a binomial. The expression we're given is X squared plus 5/7 X. Our answer choices are answer choice. A, the constant is 5/7. And then the trinomial expressed as the square of a binomial is X squared plus 5/7 X plus 5/7 equals X plus the square root of 5/7 all squared. Answer choice B the constant is 25 196th. And our trinomial and binomial are X squared plus 5/7 X plus 196th equals X-plus 5/14, all squared answer trace C. The constant is 5/14. And our trinomial and binomial are X squared plus 5/7 X plus 5/ equals X plus the square root of 70/14 all squared. In answer choice D, our constant is 196 25th and the trinomial and binomial R X squared plus 5/ X plus 196 25th equals X plus 14 5th all squared. All right. The first step in identifying the constant we need for a perfect square trinomial is to identify our B term. The B term is going to be in front of the variable to the first degree. So this case, that's an X and the B term here is 5/7. And what we're going to do to get our constant term is we're going to take our B term, divide it by two and then square it. So in this case, we're taking 5/7 dividing that by two and then squaring it. Let's do the work inside the parentheses. First, we take 5/7 divided by two. We could rewrite that as two fractions. 5/7 division symbol divided by two, which is the same as 5/ multiplied by the reciprocal of the second term one half. And so now we can do the division straight or excuse me, multiplications straight across. So multiplying our numerators five multiplied by one is 57, multiplied by two, is 14. So that is 5/7 divided by two. We get 5/14. We still need to square that. But I want to pause for a moment. This term B divided by two, that's inside of our parentheses. We will need that in a few moments. But now going back to getting our constant term, we want to square 5/14. So we need to square, both the numerator and the denominator five squared is 14 squared is 196. So our constant term is 25 196. Now writing this as a trinomial and expressing it as the square of a binomial. So we take our expression that we were given X squared plus 5/7 X. Now we add our constant term plus 196th and that is equal to. And now we put in our, and that will be adding that to our B over two or B divided by two that we identified. So X plus 5/14 closed parentheses and squaring that we look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. 32. x²+4/5x

Concetti chiave

Perfect Square Trinomial

Completing the Square

Factoring Quadratics

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