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 squared plus 𝑐. What is the value of 𝑐?
Before we can figure out what 𝑐 or 𝑏 is for that matter, we’ll need to actually complete the square for this function. The first thing I wanna do is go ahead and set our function equal to zero. So I’ve changed that 𝑓 of 𝑥 to zero.
For completing the square, we wanna take something that looks like this 𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero and turn it to 𝑥 plus 𝑏 over two squared equals 𝑏 over two squared minus 𝑐. Staring at all of those variables might make it seem more complicated than it is. But let’s walk through it together.
The first thing we’re gonna do is look at the 𝑐 variable. We’re going to take this 46 and move it to the other side of the equation. So to do that, we’ll subtract 46 from both sides. On the right side, positive 46 minus 46 equals zero. It cancels each other out. And on the left side, we’re left with negative 46. Bring down the 𝑥 squared. Bring down the 14𝑥. What we need to do now is add 𝑏 over two squared to both sides of our equation.
Our 𝑏 value is 14. We recognize this by looking at the standard form of the equation, the top line, and seeing what is the coefficient for 𝑥. So we plug in 14 over two squared and we add that to both sides. 14 divided by two is seven. So we need to take seven squared and add it to both sides, which means we’ll add 49 to both sides of our equation. And here’s what we have: 49 minus 46 equals 𝑥 squared plus 14𝑥 plus 49. Just gonna copy this down again to give us a little bit more room. We can go ahead and say 49 minus 46 equals three.
And here’s where completing the square comes in. On the right side of our equation, we’ve created a square. 𝑥 squared plus 14𝑥 plus 49 is 𝑥 plus seven squared. You can see that here. It’s 𝑥 plus 𝑏 over two, so in our case, 14 divided by two squared. Bring down the three. What we’re going to do now is try and get everything on the same side of the equation. To do that, we’ll subtract three from both sides. And now, we have 𝑥 plus seven squared minus three is equal to zero.
Now, let’s take a look at what the question was asking. Our question tells us that we arrive at the expression 𝑥 minus 𝑏 squared plus 𝑐. Have we arrived at 𝑥 minus 𝑏 squared plus 𝑐? At first glance, you might not think that we have. But there’s something we can do to make that a little bit more clear.
What if we wrote 𝑥 minus negative seven squared plus negative three? If we write our equation 𝑥 minus negative seven squared plus negative three, we have not changed the value of our expression at all. And it’s now in the format that our question was asking for it in.
And if we look at this format, we can tell that the 𝑐 is negative three. This 𝑐 value, the value that they’re looking for, will actually be the 𝑦-intercept of this function. And we found that 𝑐 equals negative three.