WEBVTT

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Hi, I'm Fiona,

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this is TMUA 2018, paper one, question 9,

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and the question asks 'find the complete
set of values of the constant C for which

083c4f3c-b1e8-4288-9839-a21e1cd2c07c-1
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the cubic equation 2X cubed minus 3 X
squared minus twelve X + C = 0 has three

083c4f3c-b1e8-4288-9839-a21e1cd2c07c-2
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distinct real solutions,'

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and here's the options to choose from
once we get into the question.

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So let's dive right in.

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When we're dealing with polynomials,
I always like to have in mind what's

e5788219-44f1-4db0-85e9-79fdc0a10f39-1
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going on graphically.

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So this is a cubic

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here.

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If I draw my axis,
I'm thinking about the coefficient of X

19ef5213-653a-48f3-b81c-23df3b820896-1
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cubed and it's positive,
which means that this cubic is going to

19ef5213-653a-48f3-b81c-23df3b820896-2
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look something like this.

8c06efbc-3868-4413-8717-bcb22c7ea92f-0
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Also looking,
I can see that C is my constant term,

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so that will always be the Y intercept.

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And then I'm thinking in order for this
equation to have three distinct real

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solutions,
that's the same as crossing the X axis

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three times.

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And so as the value of C changes,
then my graph is going to move up or down

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the Y axis.

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And so in order to keep this cubic
crossing the X axis three times,

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I need this stationary point to stay
above the X axis,

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and I need this stationary point to stay
below the X axis.

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So I'm dealing with stationary points,

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I'm going to differentiate and take it
from there.

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I'm going to set my function equal to F
of X and then I'm going to differentiate

66124bbc-7322-436d-84e9-270b0b8a638c-1
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and I'll find F-X.

64fe4cfc-3745-46e6-b4a4-ea5c2b36b708-0
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So 6 X squared minus 6 X minus 12,
and then setting that equal to 0,

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I'll be able to solve for X to give me my
X ordinates of these two stationary

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points and find out more about them.

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Because I'm setting this equal to zero,
I can divide all across by 6 straight

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away.

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That leaves me with X squared minus X - 2
= 0, which factorises nicely.

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I get a factor of X - 2 and X + 1,
and then that gives me two equations in X

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to find my X ordinates.

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So that tells me that X = 2 and X = -1.

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This stationary point is to the left of
the Y axis,

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so that's going to have an X ordinate of
-1,

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and this one is to the right of the Y
axis.

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That's going to have an X ordinate of 2.

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Now I just want to find the Y ordinates.

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So I'm going to substitute these values
of X back into my original equation to

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get the Y ordinates of these stationary
points.

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So first of all, F of -1.

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We'll start with the one on the left.

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That's going to give me -2 still a -3 and
a +12 and not forgetting my plus C.

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So that's going to be -5 + 12.

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That's a positive 7 + C.

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And that is my Y co-ordinate of this
stationary point over here.

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And then finding F of two,
I get 2 cubed which is 8 squared * 2,

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which is 16, 2 squared which is 4 * 3.

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That's going to be -12.

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And then I get a - 24 and a + 6 so -36 +
16 is going to give me a -20.

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So let's C - 20 for the Y ordinate of
this stationary point.

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So in order to keep this graph crossing
the X axis three times,

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I need 7 + C which is the Y ordinate of
this stationary point.

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I need that to always be greater than 0
so to stay above the X axis.

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And I need C - 20 to always be below the
X axis or to be less than 0.

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Tidying these inequalities up,
I guess C has to be greater than -7 and C

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has to be less than 20.

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I'm putting those together into one
inequality.

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I get that C has to be between -7 and 20.

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So that is option B for this question.

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OK,
let's just take a moment to pause and

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reflect on this question.

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So reading the question, I could think,
here's a cubic equation,

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I need to factorise this left hand side.

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But we can breathe a sigh of relief.

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We don't need to factorise this left hand
side.

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We just need to make sure that it has
three distinct real solutions.

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We don't need to find those solutions.

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So at the start of this question,
when I first thought about what was going

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on graphically and took a few seconds to
draw this sketch,

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that helped me to see that I'm dealing
with stationary points and that the

8807b044-bd14-4642-a01e-fc37d2877919-3
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question could flow from there.

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The other thing about drawing this quick
sketch of what's going on graphically is

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it shows me that I need to have an upper
bound or a highest value for C and a

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lower bound or a lowest value for C
Scanning my solutions,

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I can see that the only two options that
do that are A&B.

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So if you had started this question,
you weren't sure how to approach it,

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and you were coming to the end of the
exam, running out of time,

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then drawing a very quick sketch like
this would lead you to to narrow your

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options down to just A&B.

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And then you're left with a 50/50 chance
of getting the question right or not.

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OK,
let's just jump on GeoGebra for a bit to

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have a look at what's going on
graphically and get a more accurate graph.

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So I've got my function,
my cubic function there in green and my

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stationary points marked in green as well.

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I've got a local maximum here and a
minimum here.

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And then I've got C marked in orange
there, which is the Y intercept.

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And let's just have a look at what
happens when C changes value.

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So as C increases, it increases up to 20.

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But when it hits 20,
this station point down here hit the X

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axis, and as it goes down to -7,
this stationary point here would hit the

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X axis at -7.

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So you'd only have two distinct solutions
to that equation rather than three.

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That's why we have the,

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we're using the less than sign rather
than the less than or equal to sign in

b3807971-e37a-4ec0-9fd9-9ace4510d23b-1
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our solution,
and let me just speed this up here so

b3807971-e37a-4ec0-9fd9-9ace4510d23b-2
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that you can see it moving a bit faster.

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So as the increases and decreases within
the interval between -7 and 20,

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you can see that the graph of my function
is staying with three distinct real

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solutions,
so three times that it crosses the X axis.