### Video Transcript

Rewrite the expression π₯ squared plus 14 π₯ in the form π₯ plus π all squared plus π. And then what is the minimum value of the function π π₯ is equal to π₯ squared plus 14?

In order to rewrite our expression in the form π₯ plus π all squared plus π, what weβre actually be doing is going to be completing the square. And when completing the square, we actually have this general rule: if we have our expression in the form π₯ squared plus ππ₯, then this will be equal to π₯ plus π over two. So thatβs our coefficient of π₯ divided by two. And then thatβs all squared minus π over two β again our coefficient of π₯ halved all squared.

Okay, so letβs use this to actually rewrite our expression. So we have π₯ squared plus 14 π₯. Our π is actually gonna be positive 14. So therefore, we can actually complete the square with our expression. So therefore, this is equal to π₯ plus 14 over two all squared minus 14 over two all squared. And thatβs because again as weβve said the 14 is our coefficient of π₯.

Okay, great, so letβs tidy this up. Okay, so now that weβve tidied it up, we can say that our expression π₯ squared plus 14 π₯ rewritten in the form π₯ plus π all squared plus π is equal to π₯ plus seven all squared minus 49. Okay, great, letβs move on to the second part of the question.

The key for the second part of the question is that we want to find the minimum value of the function. To help us work out what our minimum value is going to be for our function, we can actually think of the vertex form, which is π¦ is equal to π then π₯ minus β all squared plus π. The reason why weβre using this is because this is what weβve arrived at when we completed the square.

The first part weβre gonna have a look at is the π because this is our coefficient of π₯ squared. And what this tells us is that well, if π is greater than zero, then we know that our vertex is going to be a minimum point. However, if π is less than zero, then our vertex is going to be a maximum point. And if we think about using the shape of the graph, if we had an π₯ squared graph with positive π₯ squared, then we know we get a U-shaped parabola. However, if we have a negative π₯ squared, we know that actually weβre gonna have an inverted U-shaped parabola.

Okay, great, so that in mind we know the coefficient of our π₯ squared is one. So itβs actually greater than zero. So yes, our vertex is going to be a minimum value. Okay, now, weβre gonna take a look at π because this is whatβs gonna help us actually solve the problem. π is our optimum value. So that means itβs either the minimum or maximum value. So in our case, itβs actually gonna be the minimum value because as we said before we know that the vertex of our function is actually going to be a minimum.

So therefore, if we consider our function in the form π₯ plus π squared plus π that we achieved in the first part of the question, weβre gonna get ππ₯ is equal to π₯ plus seven all squared minus 49. Then, we can say that the negative 49 is gonna be our π value.

And itβs going to be negative 49 because if we look at our vertex form, we can see that itβs plus π on the end. And this time, we have minus 49. So therefore, we know that our optimum value is going to be negative. So therefore, we can say the minimum value of our function π π₯ is equal to π₯ squared plus 14 is equal to negative 49.