Using the Discriminant - Algebra-2

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: The discriminant determines the nature of the roots:
- If $$ \Delta > 0 $$, the equation has two distinct real roots.
- If $$ \Delta = 0 $$, the equation has one real root (a repeated root).
- If $$ \Delta < 0 $$, the equation has no real roots (two complex roots).

Quadratic Formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant and Roots: $$ \Delta = b^2 - 4ac $$

Application: Used to determine the number and type of solutions for quadratic equations, which is crucial in physics and engineering problems.

Problem: Determine the nature of the roots for the quadratic equation $$ x^2 - 4x + 4 = 0 $$.

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

Validation: Substitute into the quadratic formula: $$ x = \frac{-(-4) \pm \sqrt{0}}{2 \cdot 1} = \frac{4}{2} = 2 $$ ✓

Problem: Determine the nature of the roots for the quadratic equation $$ 2x^2 + 3x + 5 = 0 $$.

Identify coefficients: $$ a = 2 $$, $$ b = 3 $$, $$ c = 5 $$
Calculate the discriminant: $$ \Delta = 3^2 - 4 \cdot 2 \cdot 5 = 9 - 40 = -31 $$
Interpret the discriminant: Since $$ \Delta < 0 $$, the equation has no real roots (two complex roots).

Validation: Substitute into the quadratic formula: $$ x = \frac{-3 \pm \sqrt{-31}}{4} $$ ✓

Step-by-Step Approach: Always identify the coefficients $$ a $$, $$ b $$, and $$ c $$ first, then calculate the discriminant.
Visual Strategy: Use a table or chart to organize the values of $$ a $$, $$ b $$, $$ c $$, and $$ \Delta $$.
Error-Proofing: Double-check the signs and arithmetic when calculating the discriminant.
Concept Reinforcement: Practice with a variety of examples to reinforce the understanding of the discriminant and its implications on the roots of the quadratic equation.