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.