Write the equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎 is not equal to zero, in the form 𝑥 minus 𝑝 squared equals 𝑞.
Here’s where we’re starting, 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. And we’re trying to convert it into the form 𝑥 minus 𝑝 squared equals 𝑞. We’re going to need to do this by completing the square. If you look at this 𝑥 term, it doesn’t have a coefficient. And that means the first step we’re going to do is divide the left equation by 𝑎. If we divide every term by 𝑎, our new equation looks like this. 𝑥 squared plus 𝑏 over 𝑎𝑥 plus 𝑐 over 𝑎 equals zero.
To complete the square, we need to move this constant to the other side of the equation. And we do that by subtracting 𝑐 over 𝑎 from both sides. 𝑥 squared plus 𝑏 over 𝑎𝑥 equals negative 𝑐 over 𝑎. When we complete this square, we take this 𝑏 term, the coefficient of our 𝑥 to the first power, and we divide it by two. 𝑏 over 𝑎 divided by two, 𝑏 over 𝑎 divided by two is the same thing as 𝑏 over 𝑎 multiplied by one-half. And that is equal to 𝑏 over two 𝑎.
We’ve now added 𝑏 over two 𝑎 squared to the left side of the equation. And that means we need to add 𝑏 over two 𝑎 squared to the right side of the equation. Because we’ve completed the square, we have 𝑥 plus 𝑏 over two 𝑎 squared on the left side. On the right side of the equation, we want to square 𝑏 over two 𝑎. Negative 𝑐 over 𝑎 doesn’t change. We end up with 𝑏 squared over two squared times 𝑎 squared. Which we can rewrite as 𝑏 squared over four 𝑎 squared.
We’re almost done, but we know that we need one term on the left side. It needs to be 𝑞. And that means we need to take negative 𝑐 over 𝑎 plus 𝑏 squared over four 𝑎 squared and rewrite it as one term. We know that to add fractions we need a common denominator. And if we multiply negative 𝑐 over 𝑎 by four 𝑎 over four 𝑎, we get a numerator of negative four 𝑎𝑐 and a denominator of four 𝑎 squared.
Once these two fractions have a common denominator, we can add their numerator. Negative four 𝑎𝑐 plus 𝑏 squared can be rewritten to say 𝑏 squared minus four 𝑎𝑐 all over four 𝑎 squared. We wanna substitute 𝑏 squared minus four 𝑎𝑐 over four 𝑎 squared in, like this.
Now we need to note something here. By completing the square, we’ve ended up with addition here. Our question wants this in the format of 𝑥 minus 𝑝. If we wanted to do that, we could say 𝑥 minus negative 𝑏 over two 𝑎. In that case, 𝑝 would be equal to negative 𝑏 over two 𝑎. And 𝑞 would be equal to 𝑏 squared minus four 𝑎𝑐 over four 𝑎 squared.
But our question hasn’t asked us to identify 𝑝 and 𝑞. It just wants this format. And so we’re gonna leave it as the simplified form 𝑥 plus 𝑏 over two 𝑎 squared equals 𝑏 squared minus four 𝑎𝑐 over four 𝑎 squared.