Discriminant

Definition

The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It is a key concept in both the study of conic sections and in solving optimization problems involving maxima and minima.

5 Must Know Facts For Your Next Test

1. The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is defined as $b^2 - 4ac$.
2. The sign of the discriminant determines the nature of the solutions to the quadratic equation: positive discriminant means two real solutions, zero discriminant means one real solution, and negative discriminant means no real solutions.
3. In the study of conic sections, the discriminant is used to classify the type of conic section (circle, ellipse, parabola, or hyperbola) based on the values of the coefficients.
4. In optimization problems, the discriminant is used to determine whether a critical point is a local maximum, local minimum, or a saddle point.
5. The discriminant is a crucial tool in both the analysis of conic sections and the solving of optimization problems involving maxima and minima.

Review Questions

• Explain how the sign of the discriminant of a quadratic equation relates to the nature of its solutions.
  • The sign of the discriminant, $b^2 - 4ac$, determines the nature of the solutions to a quadratic equation $ax^2 + bx + c = 0$. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions, only complex conjugate solutions.
• Describe how the discriminant is used to classify the type of conic section represented by the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
  • The discriminant, $B^2 - 4AC$, is used to classify the type of conic section represented by the general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. If the discriminant is positive, the conic section is a hyperbola. If the discriminant is zero, the conic section is a parabola. If the discriminant is negative, the conic section is an ellipse (including a circle as a special case).
• Explain how the discriminant can be used to determine the nature of critical points in optimization problems.
  • In optimization problems, the discriminant of the Hessian matrix, which contains the second partial derivatives of the function, can be used to determine the nature of the critical points. If the discriminant is positive, the critical point is a local minimum. If the discriminant is negative, the critical point is a local maximum. If the discriminant is zero, the critical point is a saddle point. This information is crucial in identifying the optimal solutions to the optimization problem.

Related terms

Quadratic Equation: A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Conic Sections:

The curves that result from the intersection of a plane and a cone, including circles, ellipses, parabolas, and hyperbolas.

Optimization:

The process of finding the maximum or minimum value of a function subject to certain constraints.