Question Video: Putting Quadratic Functions in Vertex Form Mathematics

Video Transcript

In completing the square for the quadratic function 𝑓 of 𝑥 equals 𝑥 squared plus 14𝑥 plus 46, you arrive at the expression 𝑥 minus 𝑏 all squared plus 𝑐. What is the value of 𝑏?

Here, we’re given a quadratic in two forms, its general or expanded form and its completed square form. These two expressions describe the same quadratic, and therefore they’re equal. We can, therefore, form an equation 𝑥 squared plus 14𝑥 plus 46 is equal to 𝑥 minus 𝑏 all squared plus 𝑐.

Now, there are two approaches that we can take to answering this question. The two approaches involve working in opposite directions. In our first approach, we’ll manipulate the expression on the right-hand side of our equation to bring it into its expanded form. Using whatever method we’re most comfortable for squaring a binomial, we see that 𝑥 minus 𝑏 all squared is equivalent to 𝑥 squared minus two 𝑏𝑥 plus 𝑏 squared. So, the expression on the right-hand side becomes 𝑥 squared minus two 𝑏𝑥 plus 𝑏 squared plus 𝑐.

As these expressions are equivalent for all values of 𝑥, we can now compare coefficients on the two sides of the equation. The coefficients of 𝑥 squared on each side are one. And then, if we compare the coefficients of 𝑥, we have 14𝑥 on the left-hand side and negative two 𝑏𝑥 on the right-hand side, giving the equation 14 equals negative two 𝑏. We can solve this equation for 𝑏 by dividing each side by negative two, giving 𝑏 equals negative seven.

Now, we have actually completed the problem because all we we’re asked for was the value of 𝑏. But suppose we’d also been asked to determine the value of 𝑐. We could do this by comparing the constant terms on the two sides of the equation. On the left, we have positive 46. And on the right, we have 𝑏 squared plus 𝑐. So, that gives us a second equation; 𝑏 squared plus 𝑐 is equal to 46. We know that 𝑏 is equal to negative seven and negative seven squared is 49. So, substituting this value into our equation, we have 49 plus 𝑐 equals 46. And subtracting 49 from each side, we find that 𝑐 is equal to negative three.

This means that the completed square or vertex form of our quadratic, using 𝑏 equals negative seven and 𝑐 equals negative three, is 𝑥 minus negative seven all squared minus three. Of course, 𝑥 minus negative seven is better written as 𝑥 plus seven. So, we can express this as 𝑥 plus seven all squared minus three.

So, that’s our first method in which we expanded to the expression on the right-hand side. In our second method, we’ll see how we can work the other way. So, we’ll start on the left-hand side and bring it into its completed square form. We already know what we’re working towards. It’s 𝑥 plus seven all squared minus three. Now, notice that the value inside the parentheses of positive seven is exactly half the coefficient of 𝑥 in our original equation. And this is no coincidence. This will always be the case. So, we begin by writing 𝑥 squared plus 14𝑥 plus 46 as 𝑥 plus seven all squared. We’ve halved the coefficient of 𝑥 to give the value inside the parentheses.

The trouble is, though, 𝑥 plus seven all squared isn’t just equivalent to 𝑥 squared plus 14𝑥; it’s equivalent to 𝑥 squared plus 14𝑥 plus 49. So, we’ve introduced an extra 49. We, therefore, need to subtract this so that the expression we have is still equivalent to 𝑥 squared plus 14𝑥. This new expression of 𝑥 plus seven all squared minus 49 is, therefore, exactly equivalent to 𝑥 squared plus 14𝑥. We also need to include the positive 46 so that the two sides of the equation are the same.

Now, that value 49 is, of course, the square of seven. So, what we’re doing is subtracting the square of the value inside our parentheses. The final step is just to simplify. We have negative 49 plus 46, which is equivalent to negative three. And so, we’ve found that this quadratic, in its completed square form, is 𝑥 plus seven all squared minus three, which is the same as we found using our first method.

Comparing this then to the given form of 𝑥 minus 𝑏 all squared plus 𝑐, we would see that positive seven is equal to negative 𝑏. Dividing or, indeed, multiplying both sides of this equation by negative one, we find that 𝑏 is equal to negative seven. Using two methods then, that’s working in both directions, we’ve found that the value of 𝑏 is negative seven.