0:00:01.290,0:00:05.658
The general strategy for solving[br]a cubic equation is to reduce it

0:00:05.658,0:00:10.026
to a quadratic and then solve[br]the quadratic by the usual means

0:00:10.026,0:00:11.846
either by Factorizing or using

0:00:11.846,0:00:18.995
the formula. A cubic equation[br]has the form a X cubed

0:00:18.995,0:00:24.710
plus BX squared plus CX plus[br]D equals not.

0:00:25.280,0:00:28.673
It must have a Turman X cubed,[br]or it wouldn't be a cubic.

0:00:29.350,0:00:33.292
But any or all of BC&D can be 0.

0:00:34.890,0:00:37.630
So for instance X cubed.

0:00:39.010,0:00:45.360
Minus six X squared plus[br]11X minus six equals note

0:00:45.360,0:00:47.265
is a cubic.

0:00:48.530,0:00:53.578
So is 4X cubed plus 57[br]equals not.

0:00:55.620,0:00:58.970
So is. X cubed

0:00:59.560,0:01:03.040
+9 X equals not.

0:01:04.950,0:01:08.770
Just as a quadratic equation may[br]have two real roots.

0:01:09.580,0:01:12.275
So a cubic equation[br]possibly has three.

0:01:13.290,0:01:17.510
But unlike a quadratic equation[br]which main have no real

0:01:17.510,0:01:21.730
solution, a cubic equation[br]always has at least one real

0:01:21.730,0:01:23.840
root. I'll explain why later.

0:01:24.740,0:01:28.592
If a cubic does have 3[br]Routes, 2 or even all three

0:01:28.592,0:01:30.197
of them may be repeated.

0:01:31.430,0:01:37.790
This gives[br]us four

0:01:37.790,0:01:39.380
possibilities.

0:01:41.410,0:01:48.960
The equation X cubed minus[br]six X squared plus 11X.

0:01:49.850,0:01:56.700
Minus 6 equals 0 Factorizes[br]2X minus one times X

0:01:56.700,0:02:02.180
minus two times X minus[br]three equals 0.

0:02:02.990,0:02:06.110
This equation has three real[br]roots, all different.

0:02:06.840,0:02:11.350
Solutions X equals 1 X equals 2[br]or X equals 3.

0:02:11.890,0:02:16.832
I'd like to show you the graph[br]of the curve Y equals X cubed

0:02:16.832,0:02:21.421
minus six X squared plus 11X[br]minus six. I'm not very good at

0:02:21.421,0:02:22.480
drawing freehand graphs.

0:02:23.010,0:02:25.140
So here's one I prepared[br]earlier.

0:02:27.980,0:02:33.146
Notice that it starts slow down[br]to the left.

0:02:33.860,0:02:37.976
Because as X gets larger,[br]negative, so does X cubed and it

0:02:37.976,0:02:41.749
finishes high to the right[br]because there's X gets large and

0:02:41.749,0:02:43.464
positive. So does X cubed.

0:02:44.420,0:02:48.944
And the curve crosses the X axis[br]three times, once where X equals

0:02:48.944,0:02:53.468
1 once, where X equals 2 and[br]once where X equals 3. This

0:02:53.468,0:02:55.208
gives us our three separate

0:02:55.208,0:03:01.856
solutions. The[br]equation X

0:03:01.856,0:03:05.790
cubed. Minus five

0:03:05.790,0:03:11.396
X squared. Plus 8X[br]minus 4 equals 0

0:03:11.396,0:03:16.805
Factorizes 2X minus one[br]times X minus two all

0:03:16.805,0:03:21.012
squared, and that is[br]equal to 0.

0:03:22.630,0:03:26.770
In this case we have. We do have[br]3 routes, but two of them are

0:03:26.770,0:03:30.082
the same. We have X minus 2[br]squared, so we only actually

0:03:30.082,0:03:35.550
have two solutions. Again, I'll[br]show you the graph of Y equals X

0:03:35.550,0:03:39.818
cubed minus five X squared plus[br]8X minus 40 equals 0.

0:03:45.470,0:03:49.502
Again, the curve starts load[br]to the left and goes high to

0:03:49.502,0:03:53.198
the right. It crosses the X[br]axis once and then just

0:03:53.198,0:03:57.902
touches it. So we have our two[br]roots, X equals 1 and X equals

0:03:57.902,0:04:01.598
2 and it touches at the[br]repeated root X equals 2.

0:04:03.220,0:04:10.692
The equation X cubed[br]minus three X squared

0:04:10.692,0:04:14.428
plus 3X minus one

0:04:14.428,0:04:20.900
equals note. The left[br]hand side factorizes to X minus

0:04:20.900,0:04:23.300
one or cubed equals not.

0:04:23.830,0:04:27.220
So there there are three[br]factors. They're all the same

0:04:27.220,0:04:30.610
and we only have a single[br]solution. X equals 1.

0:04:31.840,0:04:36.064
The corresponding curve is Y[br]equals X cubed, minus three X

0:04:36.064,0:04:37.984
squared plus 3X minus one.

0:04:38.700,0:04:41.350
And it looks like this.

0:04:44.870,0:04:47.738
As with all the Cubix I've[br]shown you so far, it starts

0:04:47.738,0:04:50.606
slow down on the left and[br]goes high up to the right.

0:04:51.960,0:04:55.964
Notice that the curve does[br]cross the X axis at the point X

0:04:55.964,0:05:00.276
equals 1, but it is also a[br]tangent. X axis is a tangent to

0:05:00.276,0:05:03.048
the curve at this point,[br]indicating the three repeated

0:05:03.048,0:05:03.356
roots.

0:05:06.800,0:05:14.780
Now look at the equation[br]X cubed plus X squared

0:05:14.780,0:05:19.950
plus X. Minus[br]3 equals 0.

0:05:21.000,0:05:23.792
This expression Factorizes 2X

0:05:23.792,0:05:31.418
minus one. X squared plus 2X[br]plus three, so we can put this

0:05:31.418,0:05:32.972
equal to 0.

0:05:35.080,0:05:40.792
The quadratic X squared +2 X +3[br]equals not has no real

0:05:40.792,0:05:44.426
solutions. So the only[br]solution to the cubic

0:05:44.426,0:05:48.962
equation is to put X minus[br]one equal to 0, giving this

0:05:48.962,0:05:51.230
single real solution X equals[br]1.

0:05:53.400,0:05:59.715
The graph Y equals X cubed plus[br]X squared plus 6 - 3 looks like

0:05:59.715,0:06:07.292
this. And you can[br]see that it only

0:06:07.292,0:06:13.774
crosses the X axis[br]in one place.

0:06:14.380,0:06:16.006
From the graphs that I've shown

0:06:16.006,0:06:19.942
you. You can see why a cubic[br]equation always has at least

0:06:19.942,0:06:20.788
one real root.

0:06:22.110,0:06:25.656
The graph either starts large,[br]negative, and finishes large

0:06:25.656,0:06:30.384
positive. If the coefficient of[br]X cubed is positive or it will

0:06:30.384,0:06:33.536
start large positive and[br]finished down here. Large

0:06:33.536,0:06:38.264
negative if the coefficient of X[br]cubed is negative, the graph of

0:06:38.264,0:06:43.780
a cubic must cross the X axis,[br]giving you one real root. So any

0:06:43.780,0:06:47.720
problem you get that involves[br]solving a cubic equation will

0:06:47.720,0:06:49.296
have a real solution.

0:06:52.720,0:06:59.056
Now let's move on to[br]how we solve cubics.

0:06:59.090,0:07:05.310
Like a quadratic, cubic should[br]always be rearranged into the

0:07:05.310,0:07:12.774
form X cubed plus BX squared[br]plus CX plus D equals 0.

0:07:13.750,0:07:20.442
The equation X squared plus 4X[br]minus one equals 6 over X is a

0:07:20.442,0:07:26.178
cubic, but I wouldn't like to[br]try and solve it in this

0:07:26.178,0:07:31.436
particular form. We need to[br]multiply through by X, giving us

0:07:31.436,0:07:36.216
X cubed plus four X squared[br]minus X equals 6.

0:07:36.740,0:07:40.940
And then we subtract 6 from both[br]sides, giving us X cubed.

0:07:41.540,0:07:48.250
Plus four X squared minus[br]X minus 6 equals 0.

0:07:49.880,0:07:54.740
When solving cubics, it[br]helps if you know or

0:07:54.740,0:07:59.600
think you know one route[br]to start with. For

0:07:59.600,0:08:02.840
instance, take the[br]equation X cubed.

0:08:04.760,0:08:11.304
Minus five X squared[br]minus 2X plus 24

0:08:11.304,0:08:18.188
equals not. Given that[br]X equals minus two is a

0:08:18.188,0:08:22.376
solution. There is a theorem[br]called the factor theorem which

0:08:22.376,0:08:26.328
I'm not going to attempt to[br]prove here that says that if X

0:08:26.328,0:08:28.152
equals minus two is a solution

0:08:28.152,0:08:35.410
of this equation. Then X[br]+2 is a

0:08:35.410,0:08:39.350
factor. Of this whole[br]expression.

0:08:40.590,0:08:47.077
This means that X cubed minus[br]five X squared minus 2X plus 24

0:08:47.077,0:08:49.572
is equal to X +2.

0:08:50.240,0:08:56.135
Times some quadratic which[br]will call X squared plus

0:08:56.135,0:08:58.100
8X Plus B.

0:08:59.220,0:09:01.196
And then all that is[br]equal to, not.

0:09:03.060,0:09:08.505
So our task now is to find A&B[br]and we do this by a process

0:09:08.505,0:09:09.594
called synthetic division.

0:09:12.440,0:09:15.646
This involves looking[br]at the coefficients of.

0:09:16.680,0:09:18.168
The original expression.

0:09:19.550,0:09:22.610
So for instance, the coefficient[br]of X cubed is one.

0:09:23.190,0:09:28.570
The coefficient of X squared[br]is minus 5. The coefficient

0:09:28.570,0:09:31.260
of X is minus 2.

0:09:32.370,0:09:34.870
And the constant is 24.

0:09:35.650,0:09:38.128
And we just right in[br]that we're

0:09:38.128,0:09:39.898
synthetically dividing[br]by minus 2.

0:09:41.620,0:09:43.060
I leave a line.

0:09:44.740,0:09:51.019
And then bring the one down one[br]times minus two is minus 2.

0:09:52.210,0:09:55.170
Minus 5 plus minus 2.

0:09:55.830,0:09:57.039
Is minus 7.

0:09:58.190,0:10:01.178
Minus Seven times minus two is

0:10:01.178,0:10:04.834
14. 14 plus minus two

0:10:04.834,0:10:08.435
is 12. 12 times minus

0:10:08.435,0:10:11.709
2. Is minus 24?

0:10:12.220,0:10:15.916
And 24 plus minus 24 is 0.

0:10:16.710,0:10:22.422
The zero tells us that X +2 is[br]indeed a factor, and the numbers

0:10:22.422,0:10:26.910
we have here give us the[br]coefficients of the quadratic. A

0:10:26.910,0:10:32.622
is equal to minus Seven and B is[br]equal to 12. So the quadratic

0:10:32.622,0:10:36.294
that we're looking for is X[br]squared minus 7X.

0:10:37.080,0:10:42.300
Plus 12. And synthetic division[br]is explained fully in the

0:10:42.300,0:10:48.413
accompanying notes. So we've[br]reduced our cubic 2X plus

0:10:48.413,0:10:55.103
two times X squared minus[br]7X plus 12 equals zero

0:10:55.103,0:11:01.793
X squared minus 7X plus[br]12 can be factorized into

0:11:01.793,0:11:08.483
X minus three times X[br]minus four. So we have

0:11:08.483,0:11:15.173
X +2 times X minus[br]three times X minus 4.

0:11:15.810,0:11:18.390
Equals 0. Giving us.

0:11:19.300,0:11:21.540
X equals minus 2.

0:11:22.690,0:11:25.900
3 or 4.

0:11:28.330,0:11:35.000
If you don't know a[br]route, it's always worth trying

0:11:35.000,0:11:40.336
a few simple values. Let's[br]solve X cubed.

0:11:40.930,0:11:41.710
Minus.

0:11:43.010,0:11:44.220
7X.

0:11:45.620,0:11:48.460
Minus 6. Equals 0.

0:11:49.040,0:11:51.128
The simplest value should try is

0:11:51.128,0:11:53.919
one. 1 - 7.

0:11:54.820,0:12:00.462
Minus 6 - 12 so that doesn't[br]work. Let's try minus 1 - 1

0:12:00.462,0:12:06.910
cubed is minus 1 + 7 - 6 is[br]0, so minus one is a route.

0:12:07.870,0:12:10.444
Which means that X plus[br]one is a factor.

0:12:12.000,0:12:18.105
If minus one is a route, we[br]can synthetically divide through

0:12:18.105,0:12:20.880
this expression by minus one.

0:12:21.680,0:12:26.289
Coefficient of X squared, sorry[br]coefficient of X cubed is one.

0:12:27.240,0:12:31.644
The coefficient of X squared is[br]0, there's no Turman X squared.

0:12:32.600,0:12:38.480
The coefficient of X is minus[br]Seven and the constant is minus

0:12:38.480,0:12:42.400
6 and were synthetically[br]dividing by minus one.

0:12:43.260,0:12:48.132
I bring the one down one times[br]minus one is minus one.

0:12:48.650,0:12:50.946
Minus one and zero is minus one.

0:12:51.650,0:12:54.032
Minus one times minus one is

0:12:54.032,0:12:58.459
one. Minus 7 Plus[br]One is minus 6.

0:12:59.510,0:13:06.478
Minus six times minus one is 6[br]- 6 at 6 is 0.

0:13:10.340,0:13:13.994
These numbers give us the[br]coefficients of the quadratic

0:13:13.994,0:13:16.430
equation, and we have X squared.

0:13:17.420,0:13:23.129
Minus X minus 6 equals[br]0 and we now need to

0:13:23.129,0:13:24.686
solve this equation.

0:13:26.400,0:13:29.732
So we have X

0:13:29.732,0:13:33.929
plus one. Times X[br]squared.

0:13:35.990,0:13:43.036
Minus X minus 6 equal to zero[br]X squared minus X minus 6,

0:13:43.036,0:13:50.082
factorizes to X minus three X[br]+2, so we have X Plus one

0:13:50.082,0:13:56.586
times X, minus three times X +2[br]equals 0, and the three

0:13:56.586,0:14:02.548
solutions to the cubic equation[br]RX equals minus 2 - 1.

0:14:03.090,0:14:04.680
Or three.

0:14:06.930,0:14:12.186
Sometimes you may be able to[br]spot a factor.

0:14:12.750,0:14:15.480
In the equation X cubed.

0:14:16.440,0:14:23.096
Minus four X squared[br]minus 9X plus 36

0:14:23.096,0:14:24.760
equals 0.

0:14:25.840,0:14:30.262
The coefficient of X squared is[br]minus four times the coefficient

0:14:30.262,0:14:31.468
of X cubed.

0:14:32.200,0:14:34.792
And the constant is minus four

0:14:34.792,0:14:37.478
times. The coefficient of X.

0:14:38.850,0:14:43.575
This means that we're going to[br]be able to take out X minus 4 as

0:14:43.575,0:14:47.040
a factor from that those two[br]terms and those two terms.

0:14:47.830,0:14:52.858
Those two terms are X squared[br]times X minus four X squared

0:14:52.858,0:14:57.886
times, X is X cubed X squared[br]times minus four is minus

0:14:57.886,0:14:59.143
four X squared.

0:15:00.360,0:15:03.090
Those two terms are minus.

0:15:04.510,0:15:08.400
9 times X minus 4.

0:15:10.400,0:15:14.564
Minus nine times X is minus[br]9X and minus nine times minus

0:15:14.564,0:15:18.381
four is plus 36 and all that[br]is equal to 0.

0:15:20.130,0:15:25.374
And we can now factorize again[br]because we have a common factor

0:15:25.374,0:15:31.055
X minus four, so we have X[br]squared minus nine times X minus

0:15:31.055,0:15:32.366
4 equals 0.

0:15:34.140,0:15:39.756
X squared minus nine is the[br]difference of two squares, so we

0:15:39.756,0:15:45.840
can write that in as X plus[br]three times X minus three times

0:15:45.840,0:15:48.180
X minus 4 equals 0.

0:15:48.820,0:15:56.090
Giving a solutions X equals[br]minus 3, three or four.

0:15:58.420,0:16:01.060
You may have noticed that in[br]each example that we've done.

0:16:02.050,0:16:07.780
Every root. There's a factor of[br]the constant term in the

0:16:07.780,0:16:11.740
original equation. For instance,[br]three and four, both divided

0:16:11.740,0:16:16.440
into 36. As long as the[br]coefficient of X cubed is one,

0:16:16.440,0:16:18.030
this must be the case.

0:16:18.780,0:16:25.286
Because. The constant is simply[br]the product of the roots 3 * 3 *

0:16:25.286,0:16:27.196
4 is equal to 36.

0:16:27.710,0:16:30.310
This gives us another possible

0:16:30.310,0:16:36.985
approach. Consider the[br]equation X cubed

0:16:36.985,0:16:40.198
minus six X

0:16:40.198,0:16:46.888
squared. Minus six X[br]minus 7 equals 0.

0:16:47.840,0:16:50.980
Now it's possible for every[br]solution to be irrational, but

0:16:50.980,0:16:54.434
if there is a rational solution,[br]then because the coefficient of

0:16:54.434,0:16:59.144
X cubed is one, it's going to be[br]an integer, and it's going to be

0:16:59.144,0:17:00.400
a factor of 7.

0:17:01.010,0:17:05.462
This only leaves us with four[br]possibilities, 1 - 1 Seven and

0:17:05.462,0:17:09.543
minus Seven, so we can try each[br]of them in turn.

0:17:10.250,0:17:14.839
You can see fairly quickly that[br]one and minus one don't work, so

0:17:14.839,0:17:15.898
let's try 7.

0:17:17.000,0:17:20.130
Rather than substituting 7 into[br]this expression and having to

0:17:20.130,0:17:24.199
workout 7 cubed and so on, what[br]I'm going to do is synthetically

0:17:24.199,0:17:29.207
divide by 7, because if Seven is[br]a route I'll end up with a 0 and

0:17:29.207,0:17:32.650
What's more, the division will[br]give me the quadratic that I'm

0:17:32.650,0:17:35.900
looking for. So let's[br]synthetically divide this

0:17:35.900,0:17:37.136
expression by 7.

0:17:38.790,0:17:40.410
The coefficient of X cubed is

0:17:40.410,0:17:43.335
one. Coefficient of X squared is

0:17:43.335,0:17:48.614
minus 6. The coefficient of X is[br]minus 6 and the constant is

0:17:48.614,0:17:51.526
minus Seven and were[br]synthetically dividing by 7.

0:17:54.600,0:18:00.102
Bring the one down 1 * 7 is[br]Seven 7 - 6 is one.

0:18:00.950,0:18:04.424
1 * 7 is 7 Seven and minus six

0:18:04.424,0:18:07.330
is one. 1 * 7 is 7.

0:18:08.130,0:18:10.470
7 and minus 70.

0:18:10.970,0:18:16.767
So 7 is indeed a route, and[br]the resulting quadratic is

0:18:16.767,0:18:20.983
X squared plus X plus one[br]equals not.

0:18:22.600,0:18:25.912
Now you'll find if you try to[br]solve it that the quadratic

0:18:25.912,0:18:27.568
equation X squared plus X plus

0:18:27.568,0:18:31.950
one equals not. Has no[br]real solutions, so the

0:18:31.950,0:18:36.800
only possible solution to[br]this cubic is X equals 7.

0:18:39.520,0:18:44.510
Certain basic identity's which[br]you may wish to learn can

0:18:44.510,0:18:49.001
help. In fact, rising both[br]cubics and quadratics. I'll

0:18:49.001,0:18:51.496
just give you 1 example.

0:18:52.730,0:18:56.700
We have the equation X cubed[br]plus three X squared.

0:18:57.530,0:19:02.270
Plus 3X plus[br]one equals 0.

0:19:04.330,0:19:11.370
1331 is the standard expansion[br]of X plus one cubed.

0:19:12.900,0:19:18.269
So the solution to this equation[br]is X plus one cubed equals 0.

0:19:18.800,0:19:23.656
We have The root minus[br]One X equals minus one

0:19:23.656,0:19:25.072
repeated three times.

0:19:27.690,0:19:34.962
If you can't find a factor[br]by these methods, then draw an

0:19:34.962,0:19:37.992
accurate graph of the cubic

0:19:37.992,0:19:41.960
expression. The points where it[br]crosses the X axis.

0:19:42.670,0:19:44.266
Will give you the solutions to

0:19:44.266,0:19:48.504
the equation. But their accuracy[br]will be limited to the accuracy

0:19:48.504,0:19:49.572
of your graph.

0:19:50.600,0:19:54.656
You might indeed find the graph[br]crosses the X axis at a point

0:19:54.656,0:19:58.400
that would suggest a factor. For[br]instance, if you draw craft a

0:19:58.400,0:20:01.520
graph that appears to cross the[br]X axis would say.

0:20:02.080,0:20:06.461
X equals 1/2. Then it's worth[br]trying to find out if X minus

0:20:06.461,0:20:08.146
1/2 is indeed a factor.

0:20:10.130,0:20:12.080
Let's look at the equation.

0:20:12.590,0:20:14.290
X cubed

0:20:15.580,0:20:18.188
Plus four X squared.

0:20:19.030,0:20:23.638
Plus X minus[br]5 equals 0.

0:20:26.900,0:20:30.090
Now this equation won't yield[br]affected by any of the methods

0:20:30.090,0:20:32.990
that I've discussed, so it's[br]time to draw a graph.

0:20:36.270,0:20:40.586
And this is the graph of Y[br]equals X cubed plus four X

0:20:40.586,0:20:42.246
squared plus 6 - 5.

0:20:44.450,0:20:49.476
It crosses the X axis at three[br]places, 1 near X equals minus 3,

0:20:49.476,0:20:51.630
one near X equals minus 2.

0:20:52.160,0:20:54.320
And one near X equals 1.

0:20:56.010,0:20:59.718
Pending on how accurate your[br]graph is, you may be able to

0:20:59.718,0:21:03.117
pinpoint it a bit closer than[br]this, and we get solutions.

0:21:03.750,0:21:07.090
X is approximately equal to

0:21:07.090,0:21:11.206
minus 3.2.[br]Minus 1.7

0:21:11.206,0:21:14.410
and not .9.

0:21:16.230,0:21:18.950
These solutions may be[br]accurate enough for your

0:21:18.950,0:21:22.350
needs, but if you require more[br]accurate answers then you

0:21:22.350,0:21:24.730
should use a numerical[br]algorithm using the

0:21:24.730,0:21:28.130
approximate answers. If you[br]obtain from the graph as a

0:21:28.130,0:21:28.810
starting point.