Video: Finding the Minimum Value of a Quadratic Expression by Completing the Square

Rewrite the expression π‘₯Β² + 14π‘₯ in the form (π‘₯ + 𝑝)Β² + π‘ž. What is the minimum value of the function 𝑓(π‘₯) = π‘₯Β² + 14?

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Video Transcript

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

In order to rewrite our expression in the form π‘₯ plus 𝑝 all squared plus π‘ž, what we’re 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 π‘Žπ‘₯, then this will be equal to π‘₯ plus π‘Ž over two. So that’s our coefficient of π‘₯ divided by two. And then that’s all squared minus π‘Ž over two β€” again our coefficient of π‘₯ halved all squared.

Okay, so let’s use this to actually rewrite our expression. So we have π‘₯ squared plus 14 π‘₯. Our π‘Ž 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 that’s because again as we’ve said the 14 is our coefficient of π‘₯.

Okay, great, so let’s tidy this up. Okay, so now that we’ve tidied it up, we can say that our expression π‘₯ squared plus 14 π‘₯ rewritten in the form π‘₯ plus 𝑝 all squared plus π‘ž is equal to π‘₯ plus seven all squared minus 49. Okay, great, let’s 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 𝑦 is equal to π‘Ž then π‘₯ minus β„Ž all squared plus π‘˜. The reason why we’re using this is because this is what we’ve arrived at when we completed the square.

The first part we’re gonna have a look at is the π‘Ž because this is our coefficient of π‘₯ squared. And what this tells us is that well, if π‘Ž is greater than zero, then we know that our vertex is going to be a minimum point. However, if π‘Ž 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 we’re gonna have an inverted U-shaped parabola.

Okay, great, so that in mind we know the coefficient of our π‘₯ squared is one. So it’s actually greater than zero. So yes, our vertex is going to be a minimum value. Okay, now, we’re gonna take a look at π‘˜ because this is what’s gonna help us actually solve the problem. π‘˜ is our optimum value. So that means it’s either the minimum or maximum value. So in our case, it’s 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 𝑝 squared plus π‘ž that we achieved in the first part of the question, we’re gonna get 𝑓π‘₯ 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 it’s going to be negative 49 because if we look at our vertex form, we can see that it’s 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 𝑓 π‘₯ is equal to π‘₯ squared plus 14 is equal to negative 49.

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