Let , be the cubic equation. By dividing the equation with we obtain:
where , , .
The equation above is called a normalized cubic equation.
The square member we remove by the substitution . Now we have:
In addition to tags and we obtain the canonical form of the cubic equation:
It is enough to solve the cubic equation of this type.
The solution of the cubic equation we search in form:
These solution must satisfy the initial equation, that is:
After transformation of the previous equation, we obtain:
We choose as an additional requirement, because each number is possible in the infinite way to display in the form of the sum of two numbers.
Therefore, we need to solve the following system of equations:
Systems of equations (1) and (2) are not equivalent. The solution of the system (1) is the solution of the system (2), however, the reversal does not have to be valid. Therefore, we choose solutions which satisfy the equation .
By the Vieta’s formulas, solutions of the system (2) are roots of the quadratic equation
By using the formula for solutions of the quadratic equation, we obtain:
Therefore, we have
The solution of the equation we write in the following form:
The formula above is called the Cardano’s formula.
The expression which appears in the Cardano’s formula is called the discriminant of the cubic equation . The discriminant of the cubic equation we will denote as .
If , then the cubic equation has one real and two complex conjugate roots; if , then the equation has three real roots, whereby at least two roots are equal; if then the equation has three distinct real roots.
Modified Cardano’s formula
be third root of 1. Then
be any value of third root and
Then the solutions of the cubic equation we can write in the form:
Example 1. Solve the following equation
The discriminant of the given equation is equal to:
Therefore, and equation has one real and two complex conjugate solutions.
By the Cardano’s formula we have:
The solutions of the given equation are:
We can use formulas above when and . When , we have a different situation, because in the Cardano’s formula appears the square root of a negative number, that is, we have complex numbers.
For example, how to solve the equation ?
By using the Cardano’s formula, we obtain:
what we can write as
If and , then
because and are complex conjugate numbers and they have the same modulus.
Now we have, , that is:
It is valid: and solutions of the equation are real numbers:
The angle we can eliminate in the following way. We know that
and we can obtain from the coefficients of the equation:
where, if then the solutions are changing the sign.
In our case:
Therefore, the one solution of the given equation is
Similar, we obtain
In general, the solutions of the cubic equation , where and , are:
where and if , then solutions are changing the sign.