Exploring Cubic Equations with One Real Root

Understanding the nature of roots in cubic equations is crucial for both mathematical theory and practical applications. A cubic equation in its general form is given by ax^3 bx^2 cx d 0, where a, b, c, d are real coefficients, and a ≠ 0. This article delves into the conditions under which a cubic equation has exactly one real root, exploring its mathematical foundations and implications.

Introduction to Cubic Equations

A cubic equation is a polynomial equation with the highest degree of three. While a general cubic equation can have up to three real roots, certain conditions can limit the number of real roots it can possess. We will focus on the scenario where there is only one real root. This situation arises in specific cases where the other roots are complex and conjugate of each other.

Conditions for a Single Real Root

For a cubic equation ax^3 bx^2 cx d 0 to have exactly one real root, the remaining roots must be either complex or conjugate pairs. We can represent such a scenario using the following steps:

Step 1: Identifying a Real Root

Assume that x a is a real root of the equation. This means that the equation can be factored as follows:

(x - a)(x - b ± ci) 0

Where b is a real number and c ≠ 0.

Step 2: Simplifying the Equation

By expanding the factored form, we get:

(x - a)(x^2 - 2bx b^2 c^2) 0

This simplifies to:

ax^3 bx^2 cx d 0

Where the quadratic equation x^2 - 2bx b^2 c^2 0 has no real roots (since its discriminant is negative).

Step 3: Condition for a Single Real Root

For the quadratic equation x^2 - 2bx b^2 c^2 0 to have no real roots, the discriminant must be less than zero:

u0394 (2b)^2 - 4(1)(b^2 c^2)

Simplifying the discriminant:

4b^2 - 4b^2 - 4c^2

-4c^2

This inequality is always true when c ≠ 0.

Implications and Applications

Understanding the conditions under which a cubic equation has exactly one real root is crucial in various mathematical and applied fields. For instance, in engineering and physics, such equations may model real-world scenarios where complex behaviors lead to specific outcomes.

Examples of Cubic Equations with One Real Root

Let's consider a specific example to illustrate the concept. Suppose we have the following cubic equation:

x^3 - 3x^2 4x - 2 0

To find the real root, we can use numerical methods or factorization techniques. Observing the equation, we can notice that x 1 is a root. This means the equation can be factored as follows:

(x - 1)(x^2 - 2x 2) 0

The quadratic equation x^2 - 2x 2 0 has no real roots, as its discriminant is negative:

u0394 (-2)^2 - 4(1)(2) 4 - 8 -4

Hence, the equation x^3 - 3x^2 4x - 2 0 has exactly one real root, which is x 1.

Conclusion

In summary, a cubic equation can have exactly one real root under specific conditions, which can be identified through factoring and examining the roots of associated quadratic equations. Understanding these conditions not only deepens our knowledge of algebraic structures but also provides valuable insights into real-world applications where complex behaviors lead to specific outcomes.