Solving Quadratic Equations by General Rule: guided by Discriminant

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Discriminant Cases in Quadratic Equations - Understanding Solution Types

1️⃣ Understand the quadratic formula and its components
2️⃣ Learn what the discriminant is and why it's important
3️⃣ Master the 4 key cases of the discriminant
4️⃣ Predict solution types without solving the equation
5️⃣ Distinguish between real, imaginary, rational, and irrational solutions

📐 The Quadratic Formula Revisited

For any quadratic equation in the form , we always have two solutions given by the quadratic formula. Understanding what determines the nature of these solutions is crucial.

The Quadratic Formula

This gives us two solutions: and

🎯 The Discriminant: The Key to Everything

The expression under the square root sign is called the discriminant. This single value determines whether our solutions will be real or imaginary, rational or irrational, equal or distinct.

The Discriminant

The Greek letter Δ (delta) represents the discriminant
This value determines the nature of both solutions!

🔍 The Four Cases of the Discriminant

Based on the value of the discriminant, we have exactly four possible cases that determine the type of solutions we'll get.

The Four Discriminant Cases

Case 1: Δ = 0 (Equal to Zero)

Condition:
Result: One repeated real root (two equal solutions)
Type: Real, rational, repeated root
The quadratic formula simplifies to:

Case 2: Δ < 0 (Negative)

Condition:
Result: Two complex (imaginary) roots
Type: Complex numbers with real and imaginary parts
Solutions have the form: and

Case 3: Δ > 0 and is a Perfect Square

Condition: (where k is an integer)
Result: Two distinct real rational roots
Type: Real, rational, distinct roots
Examples: Δ = 4, 9, 16, 25, 36, ... (perfect squares)

Case 4: Δ > 0 but Not a Perfect Square

Condition: but not a perfect square
Result: Two distinct real irrational roots
Type: Real, irrational, distinct roots
Examples: Δ = 2, 3, 5, 7, 8, 10, ... (non-perfect squares)

📝 Worked Examples

Example Problems

Example 1:

Step 1: Identify a = 1, b = -6, c = 9
Step 2: Calculate Δ =
Step 3: Since Δ = 0, we have one repeated real root
Solution: (repeated root)

Example 2:

Step 1: Identify a = 1, b = 2, c = 5
Step 2: Calculate Δ =
Step 3: Since Δ < 0, we have two complex roots
Solutions:

Example 3:

Step 1: Identify a = 1, b = -5, c = 6
Step 2: Calculate Δ =
Step 3: Since Δ = 1 (perfect square), we have two rational roots
Solutions: , so or

Example 4:

Step 1: Identify a = 1, b = -4, c = 1
Step 2: Calculate Δ =
Step 3: Since Δ = 12 (not a perfect square), we have two irrational roots
Solutions:

🧠 Quick Reference Guide

Discriminant Decision Tree

  • Δ = 0: One repeated real root (touches x-axis once)
  • Δ < 0: Two complex roots (doesn't touch x-axis)
  • Δ > 0 and perfect square: Two rational roots (crosses x-axis at rational points)
  • Δ > 0 and not perfect square: Two irrational roots (crosses x-axis at irrational points)

🎯 Key Points to Remember

  • Discriminant formula: Δ = b² - 4ac
  • Always two solutions: Every quadratic has exactly 2 solutions (counting multiplicity)
  • Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
  • Complex roots come in pairs: If one root is a + bi, the other is a - bi
  • Graph interpretation: Discriminant tells you how many x-intercepts the parabola has
  • Prediction power: You can determine solution types without solving the equation!
283. Relations and Functions
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