A cubic equation is a polynomial equation of degree 3, typically written in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a \neq 0\).
Cardano's Formula
Cardano's formula is a method for solving cubic equations. It was published by Gerolamo Cardano in 1545 and is considered one of the significant mathematical discoveries of the 16th century.
The Formula
For a cubic equation in the general form \(ax^3 + bx^2 + cx + d = 0\), Cardano's formula gives the solution:
\[ x = S + T - \frac{b}{3a} \]
Where:
\( S = \sqrt[3]{R + \sqrt{Q^3 + R^2}} \)
\( T = \sqrt[3]{R - \sqrt{Q^3 + R^2}} \)
\( Q = \frac{3ac - b^2}{9a^2} \)
\( R = \frac{9abc - 27a^2d - 2b^3}{54a^3} \)
Meaning of S, T, Q, and R
S and T: These are the two cube roots that form the basis of the solution. They represent the two parts of the solution that, when added together (along with a correction term), give one of the roots of the cubic equation.
Q: This term is related to the coefficients of the cubic equation and helps in simplifying the solution process.
R: This term, along with Q, helps determine the nature of the roots (whether they are all real, or if there are complex roots).
The quantity \(Q^3 + R^2\) is known as the discriminant of the cubic equation. Its value determines whether the equation has three real roots, or one real and two complex conjugate roots.
How S, T, Q, and R Lead to the Solution
First, calculate Q and R using the coefficients a, b, c, and d of the cubic equation.
Use Q and R to calculate the discriminant \(Q^3 + R^2\).
If \(Q^3 + R^2 > 0\), there is one real root and two complex conjugate roots.
If \(Q^3 + R^2 = 0\), all roots are real, and at least two are equal.
If \(Q^3 + R^2 < 0\), all three roots are real and unequal.
Calculate S and T using the formulas given above.
The first root is given by \(x_1 = S + T - \frac{b}{3a}\).
If there are complex roots, they are given by:
\[x_2 = -\frac{1}{2}(S+T) - \frac{b}{3a} + i\frac{\sqrt{3}}{2}(S-T)\]
\[x_3 = -\frac{1}{2}(S+T) - \frac{b}{3a} - i\frac{\sqrt{3}}{2}(S-T)\]
If all roots are real, the other two roots can be found using the relationships between roots and coefficients of the cubic equation.
Solving Steps
Ensure the cubic equation is in the standard form \(ax^3 + bx^2 + cx + d = 0\)
Calculate Q and R using the formulas provided
Determine S and T
Apply the formula \(x = S + T - \frac{b}{3a}\) to find one root
Use polynomial division or other methods to find the remaining roots
Using Cardano's formula, we can find that this equation has three real roots: 1, 2, and 3.
In this graph, the blue curve represents our cubic equation. The red dots indicate where the curve intersects the x-axis, corresponding to the roots of the equation at x = 1, 2, and 3.
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