WEBVTT

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The general strategy for solving
a cubic equation is to reduce it

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to a quadratic and then solve
the quadratic by the usual means

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either by Factorizing or using

00:00:11.846 --> 00:00:18.995
the formula. A cubic equation
has the form a X cubed

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plus BX squared plus CX plus
D equals not.

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It must have a Turman X cubed,
or it wouldn't be a cubic.

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But any or all of BC&D can be 0.

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So for instance X cubed.

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Minus six X squared plus
11X minus six equals note

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is a cubic.

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So is 4X cubed plus 57
equals not.

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So is. X cubed

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+9 X equals not.

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Just as a quadratic equation may
have two real roots.

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So a cubic equation
possibly has three.

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But unlike a quadratic equation
which main have no real

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solution, a cubic equation
always has at least one real

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root. I'll explain why later.

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If a cubic does have 3
Routes, 2 or even all three

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of them may be repeated.

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This gives
us four

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possibilities.

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The equation X cubed minus
six X squared plus 11X.

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Minus 6 equals 0 Factorizes
2X minus one times X

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minus two times X minus
three equals 0.

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This equation has three real
roots, all different.

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Solutions X equals 1 X equals 2
or X equals 3.

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I'd like to show you the graph
of the curve Y equals X cubed

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minus six X squared plus 11X
minus six. I'm not very good at

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drawing freehand graphs.

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So here's one I prepared
earlier.

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Notice that it starts slow down
to the left.

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Because as X gets larger,
negative, so does X cubed and it

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finishes high to the right
because there's X gets large and

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positive. So does X cubed.

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And the curve crosses the X axis
three times, once where X equals

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1 once, where X equals 2 and
once where X equals 3. This

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gives us our three separate

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solutions. The
equation X

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cubed. Minus five

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X squared. Plus 8X
minus 4 equals 0

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Factorizes 2X minus one
times X minus two all

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squared, and that is
equal to 0.

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In this case we have. We do have
3 routes, but two of them are

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the same. We have X minus 2
squared, so we only actually

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have two solutions. Again, I'll
show you the graph of Y equals X

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cubed minus five X squared plus
8X minus 40 equals 0.

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Again, the curve starts load
to the left and goes high to

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the right. It crosses the X
axis once and then just

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touches it. So we have our two
roots, X equals 1 and X equals

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2 and it touches at the
repeated root X equals 2.

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The equation X cubed
minus three X squared

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plus 3X minus one

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equals note. The left
hand side factorizes to X minus

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one or cubed equals not.

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So there there are three
factors. They're all the same

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and we only have a single
solution. X equals 1.

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The corresponding curve is Y
equals X cubed, minus three X

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squared plus 3X minus one.

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And it looks like this.

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As with all the Cubix I've
shown you so far, it starts

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slow down on the left and
goes high up to the right.

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Notice that the curve does
cross the X axis at the point X

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equals 1, but it is also a
tangent. X axis is a tangent to

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the curve at this point,
indicating the three repeated

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roots.

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Now look at the equation
X cubed plus X squared

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plus X. Minus
3 equals 0.

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This expression Factorizes 2X

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minus one. X squared plus 2X
plus three, so we can put this

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equal to 0.

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The quadratic X squared +2 X +3
equals not has no real

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solutions. So the only
solution to the cubic

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equation is to put X minus
one equal to 0, giving this

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single real solution X equals
1.

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The graph Y equals X cubed plus
X squared plus 6 - 3 looks like

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this. And you can
see that it only

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crosses the X axis
in one place.

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From the graphs that I've shown

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you. You can see why a cubic
equation always has at least

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one real root.

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The graph either starts large,
negative, and finishes large

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positive. If the coefficient of
X cubed is positive or it will

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start large positive and
finished down here. Large

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negative if the coefficient of X
cubed is negative, the graph of

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a cubic must cross the X axis,
giving you one real root. So any

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problem you get that involves
solving a cubic equation will

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have a real solution.

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Now let's move on to
how we solve cubics.

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Like a quadratic, cubic should
always be rearranged into the

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form X cubed plus BX squared
plus CX plus D equals 0.

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The equation X squared plus 4X
minus one equals 6 over X is a

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cubic, but I wouldn't like to
try and solve it in this

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particular form. We need to
multiply through by X, giving us

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X cubed plus four X squared
minus X equals 6.

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And then we subtract 6 from both
sides, giving us X cubed.

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Plus four X squared minus
X minus 6 equals 0.

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When solving cubics, it
helps if you know or

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think you know one route
to start with. For

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instance, take the
equation X cubed.

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Minus five X squared
minus 2X plus 24

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equals not. Given that
X equals minus two is a

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solution. There is a theorem
called the factor theorem which

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I'm not going to attempt to
prove here that says that if X

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equals minus two is a solution

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of this equation. Then X
+2 is a

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factor. Of this whole
expression.

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This means that X cubed minus
five X squared minus 2X plus 24

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is equal to X +2.

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Times some quadratic which
will call X squared plus

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8X Plus B.

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And then all that is
equal to, not.

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So our task now is to find A&B
and we do this by a process

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called synthetic division.

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This involves looking
at the coefficients of.

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The original expression.

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So for instance, the coefficient
of X cubed is one.

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The coefficient of X squared
is minus 5. The coefficient

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of X is minus 2.

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And the constant is 24.

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And we just right in
that we're

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synthetically dividing
by minus 2.

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I leave a line.

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And then bring the one down one
times minus two is minus 2.

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Minus 5 plus minus 2.

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Is minus 7.

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Minus Seven times minus two is

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14. 14 plus minus two

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is 12. 12 times minus

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2. Is minus 24?

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And 24 plus minus 24 is 0.

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The zero tells us that X +2 is
indeed a factor, and the numbers

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we have here give us the
coefficients of the quadratic. A

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is equal to minus Seven and B is
equal to 12. So the quadratic

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that we're looking for is X
squared minus 7X.

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Plus 12. And synthetic division
is explained fully in the

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accompanying notes. So we've
reduced our cubic 2X plus

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two times X squared minus
7X plus 12 equals zero

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X squared minus 7X plus
12 can be factorized into

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X minus three times X
minus four. So we have

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X +2 times X minus
three times X minus 4.

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Equals 0. Giving us.

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X equals minus 2.

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3 or 4.

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If you don't know a
route, it's always worth trying

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a few simple values. Let's
solve X cubed.

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Minus.

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7X.

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Minus 6. Equals 0.

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The simplest value should try is

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one. 1 - 7.

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Minus 6 - 12 so that doesn't
work. Let's try minus 1 - 1

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cubed is minus 1 + 7 - 6 is
0, so minus one is a route.

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Which means that X plus
one is a factor.

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If minus one is a route, we
can synthetically divide through

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this expression by minus one.

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Coefficient of X squared, sorry
coefficient of X cubed is one.

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The coefficient of X squared is
0, there's no Turman X squared.

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The coefficient of X is minus
Seven and the constant is minus

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6 and were synthetically
dividing by minus one.

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I bring the one down one times
minus one is minus one.

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Minus one and zero is minus one.

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Minus one times minus one is

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one. Minus 7 Plus
One is minus 6.

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Minus six times minus one is 6
- 6 at 6 is 0.

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These numbers give us the
coefficients of the quadratic

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equation, and we have X squared.

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Minus X minus 6 equals
0 and we now need to

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solve this equation.

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So we have X

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plus one. Times X
squared.

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Minus X minus 6 equal to zero
X squared minus X minus 6,

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factorizes to X minus three X
+2, so we have X Plus one

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times X, minus three times X +2
equals 0, and the three

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solutions to the cubic equation
RX equals minus 2 - 1.

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Or three.

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Sometimes you may be able to
spot a factor.

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In the equation X cubed.

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Minus four X squared
minus 9X plus 36

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equals 0.

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The coefficient of X squared is
minus four times the coefficient

00:14:30.262 --> 00:14:31.468
of X cubed.

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And the constant is minus four

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times. The coefficient of X.

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This means that we're going to
be able to take out X minus 4 as

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a factor from that those two
terms and those two terms.

00:14:47.830 --> 00:14:52.858
Those two terms are X squared
times X minus four X squared

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times, X is X cubed X squared
times minus four is minus

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four X squared.

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Those two terms are minus.

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9 times X minus 4.

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Minus nine times X is minus
9X and minus nine times minus

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four is plus 36 and all that
is equal to 0.

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And we can now factorize again
because we have a common factor

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X minus four, so we have X
squared minus nine times X minus

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4 equals 0.

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X squared minus nine is the
difference of two squares, so we

00:15:39.756 --> 00:15:45.840
can write that in as X plus
three times X minus three times

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X minus 4 equals 0.

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Giving a solutions X equals
minus 3, three or four.

00:15:58.420 --> 00:16:01.060
You may have noticed that in
each example that we've done.

00:16:02.050 --> 00:16:07.780
Every root. There's a factor of
the constant term in the

00:16:07.780 --> 00:16:11.740
original equation. For instance,
three and four, both divided

00:16:11.740 --> 00:16:16.440
into 36. As long as the
coefficient of X cubed is one,

00:16:16.440 --> 00:16:18.030
this must be the case.

00:16:18.780 --> 00:16:25.286
Because. The constant is simply
the product of the roots 3 * 3 *

00:16:25.286 --> 00:16:27.196
4 is equal to 36.

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This gives us another possible

00:16:30.310 --> 00:16:36.985
approach. Consider the
equation X cubed

00:16:36.985 --> 00:16:40.198
minus six X

00:16:40.198 --> 00:16:46.888
squared. Minus six X
minus 7 equals 0.

00:16:47.840 --> 00:16:50.980
Now it's possible for every
solution to be irrational, but

00:16:50.980 --> 00:16:54.434
if there is a rational solution,
then because the coefficient of

00:16:54.434 --> 00:16:59.144
X cubed is one, it's going to be
an integer, and it's going to be

00:16:59.144 --> 00:17:00.400
a factor of 7.

00:17:01.010 --> 00:17:05.462
This only leaves us with four
possibilities, 1 - 1 Seven and

00:17:05.462 --> 00:17:09.543
minus Seven, so we can try each
of them in turn.

00:17:10.250 --> 00:17:14.839
You can see fairly quickly that
one and minus one don't work, so

00:17:14.839 --> 00:17:15.898
let's try 7.

00:17:17.000 --> 00:17:20.130
Rather than substituting 7 into
this expression and having to

00:17:20.130 --> 00:17:24.199
workout 7 cubed and so on, what
I'm going to do is synthetically

00:17:24.199 --> 00:17:29.207
divide by 7, because if Seven is
a route I'll end up with a 0 and

00:17:29.207 --> 00:17:32.650
What's more, the division will
give me the quadratic that I'm

00:17:32.650 --> 00:17:35.900
looking for. So let's
synthetically divide this

00:17:35.900 --> 00:17:37.136
expression by 7.

00:17:38.790 --> 00:17:40.410
The coefficient of X cubed is

00:17:40.410 --> 00:17:43.335
one. Coefficient of X squared is

00:17:43.335 --> 00:17:48.614
minus 6. The coefficient of X is
minus 6 and the constant is

00:17:48.614 --> 00:17:51.526
minus Seven and were
synthetically dividing by 7.

00:17:54.600 --> 00:18:00.102
Bring the one down 1 * 7 is
Seven 7 - 6 is one.

00:18:00.950 --> 00:18:04.424
1 * 7 is 7 Seven and minus six

00:18:04.424 --> 00:18:07.330
is one. 1 * 7 is 7.

00:18:08.130 --> 00:18:10.470
7 and minus 70.

00:18:10.970 --> 00:18:16.767
So 7 is indeed a route, and
the resulting quadratic is

00:18:16.767 --> 00:18:20.983
X squared plus X plus one
equals not.

00:18:22.600 --> 00:18:25.912
Now you'll find if you try to
solve it that the quadratic

00:18:25.912 --> 00:18:27.568
equation X squared plus X plus

00:18:27.568 --> 00:18:31.950
one equals not. Has no
real solutions, so the

00:18:31.950 --> 00:18:36.800
only possible solution to
this cubic is X equals 7.

00:18:39.520 --> 00:18:44.510
Certain basic identity's which
you may wish to learn can

00:18:44.510 --> 00:18:49.001
help. In fact, rising both
cubics and quadratics. I'll

00:18:49.001 --> 00:18:51.496
just give you 1 example.

00:18:52.730 --> 00:18:56.700
We have the equation X cubed
plus three X squared.

00:18:57.530 --> 00:19:02.270
Plus 3X plus
one equals 0.

00:19:04.330 --> 00:19:11.370
1331 is the standard expansion
of X plus one cubed.

00:19:12.900 --> 00:19:18.269
So the solution to this equation
is X plus one cubed equals 0.

00:19:18.800 --> 00:19:23.656
We have The root minus
One X equals minus one

00:19:23.656 --> 00:19:25.072
repeated three times.

00:19:27.690 --> 00:19:34.962
If you can't find a factor
by these methods, then draw an

00:19:34.962 --> 00:19:37.992
accurate graph of the cubic

00:19:37.992 --> 00:19:41.960
expression. The points where it
crosses the X axis.

00:19:42.670 --> 00:19:44.266
Will give you the solutions to

00:19:44.266 --> 00:19:48.504
the equation. But their accuracy
will be limited to the accuracy

00:19:48.504 --> 00:19:49.572
of your graph.

00:19:50.600 --> 00:19:54.656
You might indeed find the graph
crosses the X axis at a point

00:19:54.656 --> 00:19:58.400
that would suggest a factor. For
instance, if you draw craft a

00:19:58.400 --> 00:20:01.520
graph that appears to cross the
X axis would say.

00:20:02.080 --> 00:20:06.461
X equals 1/2. Then it's worth
trying to find out if X minus

00:20:06.461 --> 00:20:08.146
1/2 is indeed a factor.

00:20:10.130 --> 00:20:12.080
Let's look at the equation.

00:20:12.590 --> 00:20:14.290
X cubed

00:20:15.580 --> 00:20:18.188
Plus four X squared.

00:20:19.030 --> 00:20:23.638
Plus X minus
5 equals 0.

00:20:26.900 --> 00:20:30.090
Now this equation won't yield
affected by any of the methods

00:20:30.090 --> 00:20:32.990
that I've discussed, so it's
time to draw a graph.

00:20:36.270 --> 00:20:40.586
And this is the graph of Y
equals X cubed plus four X

00:20:40.586 --> 00:20:42.246
squared plus 6 - 5.

00:20:44.450 --> 00:20:49.476
It crosses the X axis at three
places, 1 near X equals minus 3,

00:20:49.476 --> 00:20:51.630
one near X equals minus 2.

00:20:52.160 --> 00:20:54.320
And one near X equals 1.

00:20:56.010 --> 00:20:59.718
Pending on how accurate your
graph is, you may be able to

00:20:59.718 --> 00:21:03.117
pinpoint it a bit closer than
this, and we get solutions.

00:21:03.750 --> 00:21:07.090
X is approximately equal to

00:21:07.090 --> 00:21:11.206
minus 3.2.
Minus 1.7

00:21:11.206 --> 00:21:14.410
and not .9.

00:21:16.230 --> 00:21:18.950
These solutions may be
accurate enough for your

00:21:18.950 --> 00:21:22.350
needs, but if you require more
accurate answers then you

00:21:22.350 --> 00:21:24.730
should use a numerical
algorithm using the

00:21:24.730 --> 00:21:28.130
approximate answers. If you
obtain from the graph as a

00:21:28.130 --> 00:21:28.810
starting point.