Video: Completing Squares

Write the equation 𝑥² = 30 − 13𝑥 in the form (𝑥 − 𝑝)² = 𝑞.

03:31

Video Transcript

Write the equation 𝑥 squared equals 30 minus 13𝑥 in the form 𝑥 minus 𝑝 all squared is equal to 𝑞.

In order to solve this problem and actually write our equation in the form given, what we need to do is we actually need to complete the square. In order to complete the square, what we’re gonna think about is our expression in this form. So we have it in the form 𝑥 squared plus 𝑎𝑥. This is going to be equal to 𝑥 plus and then it’s 𝑎 over two. So we halve the coefficient of 𝑥. And then that’s all squared. And then, we subtract our 𝑎 over two — so again, our halved coefficient of 𝑥 all squared.

Okay, so if we look at our equation, we can’t actually right- we can’t use completing the square straight away because that’s actually not quite in the right form. So what we want to do first of all is rearrange it. And in order to rearrange it, what we’re gonna do is actually gonna add 13𝑥 to each side. And when we do that, we actually get 𝑥 squared plus 13 𝑥 is equal to 30. And great, we now can see this and see that actually it’s in the form that we wanted because it’s in the form 𝑥 squared plus 𝑎𝑥. And actually, our 𝑎 is equal to 13 because that’s our coefficient of 𝑥.

So what we’re now gonna do is we’re actually going to complete the square. So first of all, we’re gonna have 𝑥 plus 13 over two all squared. And the reason we have that is cause like we mentioned previously our coefficient of 𝑥, so our 𝑎, is going to be halved. So if we actually halved 13, so it gives us 𝑥 plus 13 over two all squared.

And our next stage is actually to subtract 13 over two all squared. And again, it’s 13 over two because it is the coefficient of 𝑥 halved. And if you see, that’s the same as in our example in the completing the square rule. And also, it’s the same as the one that we have in the first bracket. Okay, so great, we’ve now done that and then we say that this is all equal to 30.

Okay, so now what we’re gonna do is we can actually calculate. So by doing that, we’re gonna get 𝑥 plus 13 over two all squared minus and then it’s 169 over four. And that’s because if you’re squaring a fraction, you square the numerator and the denominator. So 13 squared is 169 and two squared is four. And this is all equal to 30.

Now, because we’re actually looking to get it in the form 𝑥 minus 𝑝 squared equals 𝑞, what we’re gonna do is actually we’re gonna add 169 over four to each side, which gives us 𝑥 plus 13 over two all squared is equal to 30 plus 169 over four. So therefore, we’re gonna have 𝑥 plus 13 over two all squared is equal to- and that what I’ve done is I’ve actually converted our 30 into 120 over four. So that’s gonna give us 120 plus 169 all over four.

So therefore, we can say that the equation 𝑥 squared equals 30 minus 13𝑥 in the form 𝑥 minus 𝑝 squared is equal to 𝑞 is 𝑥 plus 13 over two all squared is equal to 289 over four. And we’ve achieved this by completing the square.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.