Can a Cubic Equation have 2 Roots?

Answer: Yes, a cubic equation can have two roots, but one of them will be repeated (a double root).

Explanation:

A cubic equation, typically of the form [Tex]ax^3 + bx^2 + cx + d = 0[/Tex], can indeed have two roots, but the nature of these roots depends on the discriminant and coefficients. If the discriminant is positive, the cubic equation will have three distinct real roots. However, if the discriminant is zero, it will have two roots, and one of these roots will be repeated, known as a double root.

Possibility of Two Roots

Contrary to common perception, a cubic equation can indeed have two roots. However, it’s important to note that one of these roots will be repeated, resulting in what is known as a double root. We explore the conditions under which a cubic equation exhibits this behavior and provide an illustrative example for clarity.

Example – Can a Cubic Equation have 2 Roots?

consider the cubic equation [Tex]x^3 − 6x^2 + 9x = 0[/Tex]. Factoring it, we get [Tex]x(x − 3)^2 = 0[/Tex], revealing that [Tex]x = 0[/Tex] and [Tex]x = 3[/Tex] are the roots. Here, [Tex]x = 3 [/Tex]is a repeated root, indicating a double root.

So, while a cubic equation can have two roots, it does so when one of those roots is repeated, resulting in a total of two distinct solutions.

Conclusion

In conclusion, while it may seem counterintuitive, a cubic equation can indeed have two roots. However, this occurs when one of the roots is repeated, resulting in a total of two distinct solutions. Understanding the nature of roots in cubic equations enhances our comprehension of polynomial functions and their properties, contributing to the broader landscape of algebra and mathematics.

Related Resources:

• Solving Cubic Equations: Definitions, Methods and Examples
• Real-Life Applications of Cube Root