Solutions of the cubic Function
We have already discussed most of it in our polynomials discussion.
The following discussion takes care of all possible values of the solution when we are facing a problem involving the cubic function.
Given:
Let’s depress the equation.
We apply the substitution
We simplify:
We factor like terms and eliminate cancelling terms:
Dividing by we can let:
and
We get the deprecated function.
Please note that when , we can simply deduct the 3 values of from those of .
This situation is ideal because the term in has vanished.
Let:
We get
We can see that :
and will make our equation .
Let’s cube the first line:
Now let:
and
We get:
To solve this we’ll use:
If , the equation has one real root and 2 complex roots
If , the equation has one root and a double root
If , the equation has three real roots
Case
We have complex roots:
with
If
We get:
Case
if
if
Case
This means:
Meaning
We get the following solutions:
We use
to get
Case
We proceed as follows:
When we replace in our equation:
By simple division we get:
Let’s simplify the left side:
From this we get:
If we let :
From our main equation:
Replacing from above
The same technic we used earlier helps us get
With
The simplified Equation is:
Remember
with
Our Solutions:
We can now find .
Case
when
Here we say:
The same transformations yield:
And:
when
Here we say:
The same transformations yield:
And:
General rule for all solutions including the complex conjugates:
From
With:
First case: when
Let’s calculate and
Now:
The values:
For values we just apply
Second case: when
Let’s calculate and
Now:
The values:
For values we just apply
Example 1:
Find the solutions of the following equation:
Solution:
Instead of using multiple tries between 72 and 10 and their factors, we are going to use the methods learned here.
To avoid big numbers, let’s divide the function by 10.
We get
We have the following coefficients:
Let’s calculate and
Both values were left at their fractions forms to avoid accuracy loss.
Now:
We are in the case where
We call our method to get the 3 real roots:
in degrees
The solutions:
Finally the solutions are:
Answer: , and
Example 2:
Find the solutions of the following equation:
Solution:
Instead of using multiple tries between 4 and 50 and their factors, we are going to use the methods learned here.
To avoid big numbers, let’s divide the function by 50.
In future exercises, we’ll use simplest methods.
We get:
We have the following coefficients:
Let’s calculate and
We plug in:
The exponent vanishes
Here:
Let’s check on
We are in the case where
We get the following solutions:
Finally the solutions are:
Answer: , and
Example 3:
Find the solutions of the following equation:
Solution:
Instead of using multiple tries between 6 and 1 and their factors, we are going to use the methods learned here.
In future exercises, we’ll use simplest methods.
We have the following coefficients:
Let’s calculate and
We plug in:
If , the equation has one real root and 2 complex roots
Let’s calculate and
Now:
The values:
For the values:
We get:
Finally the solutions are:
Answer: , and
Atlernate methods using division
The following documents show how the division can get the factors, reducing the cubic function to a product that we normally solve:
Example 1:
Find the solutions of the following equation:
Solution:
[accordion hideSpeed=”300″ showSpeed=”400″][accordion hideSpeed=”300″ showSpeed=”400″]
[item title=”Click here to see the solution of: “]
{aridoc engine=”pdfjs” width=”100%” height=”800″}images/Alternate2.pdf{/aridoc}
[/item] [/accordion]
Example 2:
Find the solutions of the following equation:
Solution:
[accordion hideSpeed=”300″ showSpeed=”400″][accordion hideSpeed=”300″ showSpeed=”400″]
[item title=”Click here to see the solution of:“]
{aridoc engine=”pdfjs” width=”100%” height=”800″}images/Alternate1.pdf{/aridoc}
[/item] [/accordion]
Example 3:
Find the solutions of the following equation:
Solution:
[accordion hideSpeed=”300″ showSpeed=”400″][accordion hideSpeed=”300″ showSpeed=”400″]
[item title=”Click here to see the solution of:“]
{aridoc engine=”pdfjs” width=”100%” height=”800″}images/Alternate3.pdf{/aridoc}
[/item] [/accordion]
HELPER FUNCTION
Open this using the right box with pointing arrow and select ‘OPEN WITH GOOGLE SHEETS’ to solve more cubic equations.
{aridoc engine=”google” width=”100%” height=”800″}images/cubic1.xlsx{/aridoc}
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