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.