The general strategy for solving
a cubic equation is to reduce it
to a quadratic and then solve
the quadratic by the usual means
either by Factorizing or using
the formula. A cubic equation
has the form a X cubed
plus BX squared plus CX plus
D equals not.
It must have a Turman X cubed,
or it wouldn't be a cubic.
But any or all of BC&D can be 0.
So for instance X cubed.
Minus six X squared plus
11X minus six equals note
is a cubic.
So is 4X cubed plus 57
equals not.
So is. X cubed
+9 X equals not.
Just as a quadratic equation may
have two real roots.
So a cubic equation
possibly has three.
But unlike a quadratic equation
which main have no real
solution, a cubic equation
always has at least one real
root. I'll explain why later.
If a cubic does have 3
Routes, 2 or even all three
of them may be repeated.
This gives
us four
possibilities.
The equation X cubed minus
six X squared plus 11X.
Minus 6 equals 0 Factorizes
2X minus one times X
minus two times X minus
three equals 0.
This equation has three real
roots, all different.
Solutions X equals 1 X equals 2
or X equals 3.
I'd like to show you the graph
of the curve Y equals X cubed
minus six X squared plus 11X
minus six. I'm not very good at
drawing freehand graphs.
So here's one I prepared
earlier.
Notice that it starts slow down
to the left.
Because as X gets larger,
negative, so does X cubed and it
finishes high to the right
because there's X gets large and
positive. So does X cubed.
And the curve crosses the X axis
three times, once where X equals
1 once, where X equals 2 and
once where X equals 3. This
gives us our three separate
solutions. The
equation X
cubed. Minus five
X squared. Plus 8X
minus 4 equals 0
Factorizes 2X minus one
times X minus two all
squared, and that is
equal to 0.
In this case we have. We do have
3 routes, but two of them are
the same. We have X minus 2
squared, so we only actually
have two solutions. Again, I'll
show you the graph of Y equals X
cubed minus five X squared plus
8X minus 40 equals 0.
Again, the curve starts load
to the left and goes high to
the right. It crosses the X
axis once and then just
touches it. So we have our two
roots, X equals 1 and X equals
2 and it touches at the
repeated root X equals 2.
The equation X cubed
minus three X squared
plus 3X minus one
equals note. The left
hand side factorizes to X minus
one or cubed equals not.
So there there are three
factors. They're all the same
and we only have a single
solution. X equals 1.
The corresponding curve is Y
equals X cubed, minus three X
squared plus 3X minus one.
And it looks like this.
As with all the Cubix I've
shown you so far, it starts
slow down on the left and
goes high up to the right.
Notice that the curve does
cross the X axis at the point X
equals 1, but it is also a
tangent. X axis is a tangent to
the curve at this point,
indicating the three repeated
roots.
Now look at the equation
X cubed plus X squared
plus X. Minus
3 equals 0.
This expression Factorizes 2X
minus one. X squared plus 2X
plus three, so we can put this
equal to 0.
The quadratic X squared +2 X +3
equals not has no real
solutions. So the only
solution to the cubic
equation is to put X minus
one equal to 0, giving this
single real solution X equals
1.
The graph Y equals X cubed plus
X squared plus 6 - 3 looks like
this. And you can
see that it only
crosses the X axis
in one place.
From the graphs that I've shown
you. You can see why a cubic
equation always has at least
one real root.
The graph either starts large,
negative, and finishes large
positive. If the coefficient of
X cubed is positive or it will
start large positive and
finished down here. Large
negative if the coefficient of X
cubed is negative, the graph of
a cubic must cross the X axis,
giving you one real root. So any
problem you get that involves
solving a cubic equation will
have a real solution.
Now let's move on to
how we solve cubics.
Like a quadratic, cubic should
always be rearranged into the
form X cubed plus BX squared
plus CX plus D equals 0.
The equation X squared plus 4X
minus one equals 6 over X is a
cubic, but I wouldn't like to
try and solve it in this
particular form. We need to
multiply through by X, giving us
X cubed plus four X squared
minus X equals 6.
And then we subtract 6 from both
sides, giving us X cubed.
Plus four X squared minus
X minus 6 equals 0.
When solving cubics, it
helps if you know or
think you know one route
to start with. For
instance, take the
equation X cubed.
Minus five X squared
minus 2X plus 24
equals not. Given that
X equals minus two is a
solution. There is a theorem
called the factor theorem which
I'm not going to attempt to
prove here that says that if X
equals minus two is a solution
of this equation. Then X
+2 is a
factor. Of this whole
expression.
This means that X cubed minus
five X squared minus 2X plus 24
is equal to X +2.
Times some quadratic which
will call X squared plus
8X Plus B.
And then all that is
equal to, not.
So our task now is to find A&B
and we do this by a process
called synthetic division.
This involves looking
at the coefficients of.
The original expression.
So for instance, the coefficient
of X cubed is one.
The coefficient of X squared
is minus 5. The coefficient
of X is minus 2.
And the constant is 24.
And we just right in
that we're
synthetically dividing
by minus 2.
I leave a line.
And then bring the one down one
times minus two is minus 2.
Minus 5 plus minus 2.
Is minus 7.
Minus Seven times minus two is
14. 14 plus minus two
is 12. 12 times minus
2. Is minus 24?
And 24 plus minus 24 is 0.
The zero tells us that X +2 is
indeed a factor, and the numbers
we have here give us the
coefficients of the quadratic. A
is equal to minus Seven and B is
equal to 12. So the quadratic
that we're looking for is X
squared minus 7X.
Plus 12. And synthetic division
is explained fully in the
accompanying notes. So we've
reduced our cubic 2X plus
two times X squared minus
7X plus 12 equals zero
X squared minus 7X plus
12 can be factorized into
X minus three times X
minus four. So we have
X +2 times X minus
three times X minus 4.
Equals 0. Giving us.
X equals minus 2.
3 or 4.
If you don't know a
route, it's always worth trying
a few simple values. Let's
solve X cubed.
Minus.
7X.
Minus 6. Equals 0.
The simplest value should try is
one. 1 - 7.
Minus 6 - 12 so that doesn't
work. Let's try minus 1 - 1
cubed is minus 1 + 7 - 6 is
0, so minus one is a route.
Which means that X plus
one is a factor.
If minus one is a route, we
can synthetically divide through
this expression by minus one.
Coefficient of X squared, sorry
coefficient of X cubed is one.
The coefficient of X squared is
0, there's no Turman X squared.
The coefficient of X is minus
Seven and the constant is minus
6 and were synthetically
dividing by minus one.
I bring the one down one times
minus one is minus one.
Minus one and zero is minus one.
Minus one times minus one is
one. Minus 7 Plus
One is minus 6.
Minus six times minus one is 6
- 6 at 6 is 0.
These numbers give us the
coefficients of the quadratic
equation, and we have X squared.
Minus X minus 6 equals
0 and we now need to
solve this equation.
So we have X
plus one. Times X
squared.
Minus X minus 6 equal to zero
X squared minus X minus 6,
factorizes to X minus three X
+2, so we have X Plus one
times X, minus three times X +2
equals 0, and the three
solutions to the cubic equation
RX equals minus 2 - 1.
Or three.
Sometimes you may be able to
spot a factor.
In the equation X cubed.
Minus four X squared
minus 9X plus 36
equals 0.
The coefficient of X squared is
minus four times the coefficient
of X cubed.
And the constant is minus four
times. The coefficient of X.
This means that we're going to
be able to take out X minus 4 as
a factor from that those two
terms and those two terms.
Those two terms are X squared
times X minus four X squared
times, X is X cubed X squared
times minus four is minus
four X squared.
Those two terms are minus.
9 times X minus 4.
Minus nine times X is minus
9X and minus nine times minus
four is plus 36 and all that
is equal to 0.
And we can now factorize again
because we have a common factor
X minus four, so we have X
squared minus nine times X minus
4 equals 0.
X squared minus nine is the
difference of two squares, so we
can write that in as X plus
three times X minus three times
X minus 4 equals 0.
Giving a solutions X equals
minus 3, three or four.
You may have noticed that in
each example that we've done.
Every root. There's a factor of
the constant term in the
original equation. For instance,
three and four, both divided
into 36. As long as the
coefficient of X cubed is one,
this must be the case.
Because. The constant is simply
the product of the roots 3 * 3 *
4 is equal to 36.
This gives us another possible
approach. Consider the
equation X cubed
minus six X
squared. Minus six X
minus 7 equals 0.
Now it's possible for every
solution to be irrational, but
if there is a rational solution,
then because the coefficient of
X cubed is one, it's going to be
an integer, and it's going to be
a factor of 7.
This only leaves us with four
possibilities, 1 - 1 Seven and
minus Seven, so we can try each
of them in turn.
You can see fairly quickly that
one and minus one don't work, so
let's try 7.
Rather than substituting 7 into
this expression and having to
workout 7 cubed and so on, what
I'm going to do is synthetically
divide by 7, because if Seven is
a route I'll end up with a 0 and
What's more, the division will
give me the quadratic that I'm
looking for. So let's
synthetically divide this
expression by 7.
The coefficient of X cubed is
one. Coefficient of X squared is
minus 6. The coefficient of X is
minus 6 and the constant is
minus Seven and were
synthetically dividing by 7.
Bring the one down 1 * 7 is
Seven 7 - 6 is one.
1 * 7 is 7 Seven and minus six
is one. 1 * 7 is 7.
7 and minus 70.
So 7 is indeed a route, and
the resulting quadratic is
X squared plus X plus one
equals not.
Now you'll find if you try to
solve it that the quadratic
equation X squared plus X plus
one equals not. Has no
real solutions, so the
only possible solution to
this cubic is X equals 7.
Certain basic identity's which
you may wish to learn can
help. In fact, rising both
cubics and quadratics. I'll
just give you 1 example.
We have the equation X cubed
plus three X squared.
Plus 3X plus
one equals 0.
1331 is the standard expansion
of X plus one cubed.
So the solution to this equation
is X plus one cubed equals 0.
We have The root minus
One X equals minus one
repeated three times.
If you can't find a factor
by these methods, then draw an
accurate graph of the cubic
expression. The points where it
crosses the X axis.
Will give you the solutions to
the equation. But their accuracy
will be limited to the accuracy
of your graph.
You might indeed find the graph
crosses the X axis at a point
that would suggest a factor. For
instance, if you draw craft a
graph that appears to cross the
X axis would say.
X equals 1/2. Then it's worth
trying to find out if X minus
1/2 is indeed a factor.
Let's look at the equation.
X cubed
Plus four X squared.
Plus X minus
5 equals 0.
Now this equation won't yield
affected by any of the methods
that I've discussed, so it's
time to draw a graph.
And this is the graph of Y
equals X cubed plus four X
squared plus 6 - 5.
It crosses the X axis at three
places, 1 near X equals minus 3,
one near X equals minus 2.
And one near X equals 1.
Pending on how accurate your
graph is, you may be able to
pinpoint it a bit closer than
this, and we get solutions.
X is approximately equal to
minus 3.2.
Minus 1.7
and not .9.
These solutions may be
accurate enough for your
needs, but if you require more
accurate answers then you
should use a numerical
algorithm using the
approximate answers. If you
obtain from the graph as a
starting point.