Cubic Equation Solver
Solving Cubic Equations
A cubic equation is a third-degree polynomial equation of the form \(ax^3 + bx^2 + cx + d = 0\), where 'a' is not zero. The solutions to this equation are called its roots.
Roots and Nature
A cubic equation always has three roots, according to the fundamental theorem of algebra. The nature of these roots is determined by the discriminant (\(\Delta\)):
- If \(\Delta > 0\), there are 3 distinct real roots.
- If \(\Delta = 0\), there are 3 real roots, with at least two being equal.
- If \(\Delta < 0\), there is 1 real root and 2 complex conjugate roots.
Key Points on the Graph
The graph of a cubic function has several important points:
Roots (x-intercepts): Where the graph crosses the x-axis.
Local Extrema: The local maximum and minimum points, found by solving \(f'(x) = 3ax^2+2bx+c = 0\).
Inflection Point: The point where the curve changes concavity, found by solving \(f''(x) = 6ax+2b = 0\).
Solves \(ax^3 + bx^2 + cx + d = 0\) and graphs the function.
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