Tough Completing The Square for Paper 1 Non Calculator | Higher GCSE Maths 14th May 2026 | REVISE9

Added: 2026-05-15

[00:00:00]Okay. So, in this question, we're going to look at completing the square.

[00:00:04]We're going to be looking at completing the square for a curve where the coefficient of x squared is not equal to one.

[00:00:13]And not just a number greater than one.

[00:00:14]In this time, it's going to be a negative coefficient.

[00:00:19]So, we have, find the coordinates of the turning point on the curve with the equation y = 9 + 18x 3x squared.

[00:00:29]And you must show your working.

[00:00:32]You need to know completing the square for this, of course. So, when we are completing the square where the coefficient is not one for x squared, the first thing we need to do is take the expression and we need to factorize out whatever number is in front of x squared, whatever that coefficient is.

[00:00:50]Now, in this case, the coefficient is -3.

[00:00:54]So, we need to factorize -3 out of this expression.

[00:00:59]Now, when we do that, nine turns into -3.

[00:01:07]Positive 18x turns into now negative 18 / -3 would be -6x.

[00:01:17]And our -3 in front of the x squared turns into positive one, -3 / -3, so we just get plus x squared.

[00:01:28]Now, I was as it was a negative quadratic, it's written in this sort of I guess you call it a sort of backwards order.

[00:01:34]So, we want to rearrange them back into the normal order that we'll see a quadratic, so x squared at the start, -6x, -3.

[00:01:46]Now, at this point, there are two ways of taking the next step. What we can do is introduce a second set of brackets, because when we complete the square, we obviously introduce a set of brackets.

[00:01:58]But it does start to get quite messy.

[00:02:00]So, what I do, it's just my personal preference, is I just extract that quadratic, x squared minus 6x minus 3, and just complete the square for this.

[00:02:11]And then we'll go back to its original.

[00:02:15]So, when we complete the square, we halve the coefficient of x. So, x minus 3 in brackets squared.

[00:02:22]When you expand this double bracket, that will give you plus nine at the end.

[00:02:27]We don't want plus nine at the end. Our quadratic has negative three.

[00:02:33]So, to get from nine to negative three, we would have to subtract 12.

[00:02:39]And that is now written in completed square form.

[00:02:43]Now, all of that completed square form is what needs to go in the bracket here.

[00:02:50]But, of course, two sets of brackets, it starts to look quite messy.

[00:02:55]So, what can be done is we can turn these curved brackets into a squared bracket.

[00:03:01]No difference with a curved bracket, just a way of differentiating which bracket matches up to which.

[00:03:06]So, we have minus three in our squared bracket, we're going to put that completed square form that we've just done.

[00:03:17]So, in brackets, x minus 3 squared minus 12.

[00:03:24]Now, you don't have to do that. It's the correct way of writing it, but what we could do, and what we need to do now, which is just to multiply everything by negative three, we could, I guess, write this up here.

[00:03:35]We could just write a Okay, we need to multiply this by negative three now. And you could just write it out there.

[00:03:42]But, I'm just going to times this by negative three. The step is relatively nice to do.

[00:03:46]It's nice to do because we don't actually have to expand this double bracket.

[00:03:52]It just gets multiplied by -3, which means -3 is on the outside.

[00:03:58]What we do need to make sure we're careful of is this -3 multiplied by -12.

[00:04:04]Negative times a negative is going to give us a positive answer, and 3 * 12 is 36. So, we get +36.

[00:04:12]And again, we'd got the same thing had we have just done that by timesing by -3. We'd have just written -3 x - 3 squared + 36.

[00:04:25]Now, from this completed square form we can just read the coordinates of the turning point. The x coordinate is here in our bracket, but we change the sign.

[00:04:36]So, that becomes positive 3 as our x coordinate.

[00:04:41]And the y coordinate is just this number at the end, positive 36.

[00:04:46]So, 36, and we don't change the sign for that one. Only when it's in brackets.

[00:04:51]So, there are the coordinates of the turning point of the curve, 3 and 36. And we got there by using completed square form. Quite a tricky version, but there we go. There is completing the square where the coefficient of x squared wasn't one, it was actually a negative number. And of course, doing all of that without a calculator.

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