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.24. x² + 4x

Accepted Answer

Hello, today we're gonna be using completing the square to identify the constant needed to transform the original binomial into a perfect square trinomial. We're then going to factor that perfect square trinomial into a square of a binomial. So what we are given is X squared plus 60 X. Now this is a binomial in the form of A X squared plus or minus BX. And if we want to transform this into a perfect square trinomial using completing the square, completing the square, asks us to find a third term C and C is going to equal to B divided by two, that quantity squared. So if we want to complete the square on X squared plus 60 X, we're going to look for a third term C and that is going to equal to 60 divided by two, that quantity squared. Now 60 divided by two is going to simplify to give us 30. And what we are left with is 30 squared and 30 squared is going to simplify to give us 900. So 900 is going to be the constant needed to transform the given binomial into a perfect square trinomial. And the form of that perfect square trinomial is just going to be what is given to us plus the new constant. So it's going to be X squared plus 60 X plus 900. So what we have here is a perfect square trinomial. And if we wanted to factor this, we need to go ahead and think to ourselves, what are two numbers that can multiply together to give us 900 but can add up to give us the middle coefficient 60. While those two numbers are going to be 30 30 30 times 30 is going to give us 930 plus 30 is going to give us 60. So the factored form of the perfect square trinomial is going to be X plus 30 times X plus 30. And this itself can be combined into a single binomial and that can be rewritten as X plus 30 squared, which is going to be the factored form of the original trinomial. So just to summarize 900 is the coefficient needed to rewrite the original binomial as a perfect square trinomial. This is going to be the form of the perfect square trinomial and this perfect square trinomial is going to factor into this squared binomial. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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.24. x² + 4x

Key Concepts

Perfect Square Trinomial

Completing the Square

Factoring Quadratics

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