QF’s Discriminant

Algebra-1

1. Fundamental Concepts

Definition: The discriminant of a quadratic equation $$ax^2 + bx + c = 0$$ is given by $$\Delta = b^2 - 4ac$$ .
Nature of Roots: Determines the nature of the roots (real, complex, distinct, or repeated).
Conditions:
- If $$\Delta > 0$$ , there are two distinct real roots.
- If $$\Delta = 0$$ , there is exactly one real root (a repeated root).
- If $$\Delta < 0$$ , there are two complex conjugate roots

2. Key Concepts

Basic Rule: $$\Delta = b^2 - 4ac$$

Degree Preservation: The discriminant helps in determining the number and type of solutions without solving the equation.

Application: Used to analyze the roots of quadratic equations in various fields such as physics and engineering.

3. Examples

Example 1 (Basic)

Problem: Determine the nature of the roots for the quadratic equation $$2x^2 + 3x + 1 = 0$$ .

Step-by-Step Solution:

Identify coefficients: $$a = 2$$ , $$b = 3$$ , $$c = 1$$ .
Calculate the discriminant: $$\Delta = b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot 1 = 9 - 8 = 1$$ .
Since $$\Delta > 0$$ , there are two distinct real roots.

Validation: Substitute into the quadratic formula to confirm the roots are real and distinct.

Example 2 (Intermediate)

Problem: For the quadratic equation $$x^2 - 4x + 4 = 0$$ , determine the nature of the roots.

Step-by-Step Solution:

Identify coefficients: $$a = 1$$ , $$b = -4$$ , $$c = 4$$ .
Calculate the discriminant: $$\Delta = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0$$ .
Since $$\Delta = 0$$ , there is exactly one real root (a repeated root).

Validation: Substitute into the quadratic formula to confirm the root is a repeated real root.

4. Problem-Solving Techniques

Visual Strategy: Use a flowchart to decide the nature of the roots based on the value of the discriminant.
Error-Proofing: Always double-check the values of $$a$$ , $$b$$ , and $$c$$ before calculating the discriminant.
Concept Reinforcement: Practice with a variety of quadratic equations to reinforce understanding of the discriminant’s role.