How to Solve Third Degree Equation
The third degree equation, also known as the cubic equation, is a polynomial equation of the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and a ≠ 0. Solving cubic equations can be challenging, but there are several methods available to find the roots of such equations. In this article, we will explore some of the most common techniques for solving third degree equations.
One of the most popular methods for solving cubic equations is the Cardano’s method, which was developed by the Italian mathematician Gerolamo Cardano in the 16th century. This method involves finding two real roots and one complex root of the equation. Here are the steps to solve a cubic equation using Cardano’s method:
1. Make a change of variables to eliminate the quadratic term: Let x = y – b/(3a), where b is the coefficient of the x^2 term and a is the coefficient of the x^3 term. This transformation will result in a new equation of the form y^3 + Py + Q = 0.
2. Solve the reduced cubic equation: Find the roots of the new equation using the formulas:
y1 = (-1/3) (P + √(P^2 – 4Q)) + (-1/3) (P – √(P^2 – 4Q))
y2 = (-1/3) (P + √(P^2 – 4Q)) – (-1/3) (P – √(P^2 – 4Q))
y3 = -y1 – y2
3. Convert the roots back to the original variable: Substitute y1, y2, and y3 back into the original equation to find the roots of the cubic equation in terms of x.
Another method for solving cubic equations is the trigonometric method, which is based on the substitution x = 2t/3. This method is particularly useful when the cubic equation has a special form, such as x^3 + px + q = 0. Here are the steps to solve a cubic equation using the trigonometric method:
1. Make the substitution x = 2t/3 and simplify the equation to get a new equation in terms of t.
2. Solve the new equation for t using the formulas:
t = (1/3) (cos(θ) + √3 sin(θ)) or t = (1/3) (cos(θ) – √3 sin(θ))
where θ is the angle whose cosine is the real part of the cube root of the discriminant (D = -4p^3 – 27q^2).
3. Convert the roots back to the original variable: Substitute t back into the original equation to find the roots of the cubic equation in terms of x.
Finally, the rational root theorem can be used to find rational roots of cubic equations. This method involves checking possible rational roots (factors of the constant term divided by factors of the leading coefficient) and using synthetic division to test them. If a rational root is found, the cubic equation can be factored into a quadratic and linear equation, which can then be solved using standard techniques.
In conclusion, solving third degree equations can be done using various methods, such as Cardano’s method, the trigonometric method, and the rational root theorem. Each method has its own advantages and is suitable for different types of cubic equations. By understanding these methods, one can effectively solve cubic equations and gain a deeper understanding of polynomial functions.
