**ERIC Number:**EJ875531

**Record Type:**Journal

**Publication Date:**2008

**Pages:**7

**Abstractor:**As Provided

**Reference Count:**0

**ISBN:**N/A

**ISSN:**ISSN-0740-8404

The Number of Real Roots of a Cubic Equation

Kavinoky, Richard; Thoo, John B.

AMATYC Review, v29 n2 p2-8 Spr 2008

To find the number of distinct real roots of the cubic equation (1) x[caret]3 + bx[caret]2 + cx + d = 0, we could attempt to solve the equation. Fortunately, it is easy to tell the number of distinct real roots of (1) without having to solve the equation. The key is the discriminant. The discriminant of (1) appears in Cardan's (or Cardano's) cubic formula. However, few students today are even aware of the cubic formula, let alone have seen it. We show how a student may come up with or be led to the discriminant of (1) without appealing to Cardan's cubic formula using ideas from a first calculus course--derivative, critical point, local extrema, and graphing--in an intuitive way. We also show how the discriminant defines a boundary in the plane across which the number of real roots of (1) changes, and apply the discriminant to determining the number of normals to the parabola y = x[caret]2 through a given point and the number of equilibrium solutions of dx/dt = (R-Rc)x-ax[caret]3, where Rc and a are positive constants and R is a parameter.

Descriptors: Calculus, Mathematics Instruction, Equations (Mathematics), Mathematical Concepts, Teaching Methods

American Mathematical Association of Two-Year Colleges. 5983 Macon Cove, Memphis, TN 38134. Tel: 901-333-4643; Fax: 901-333-4651; e-mail: amatyc@amatyc.org; Web site: http://www.amatyc.org

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A