Solving Quadratic Equations by General Rule: guided by Discriminant

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 $ax^2 + bx + c = 0$ , we always have two solutions given by the quadratic formula. Understanding what determines the nature of these solutions is crucial.

The Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This gives us two solutions:

x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

and

x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

🎯 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

\Delta = b^2 - 4ac

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:

b^2 - 4ac = 0

Result: One repeated real root (two equal solutions)

Type: Real, rational, repeated root

The quadratic formula simplifies to:

x = \frac{-b}{2a}

Case 2: Δ < 0 (Negative)

Condition:

b^2 - 4ac < 0

Result: Two complex (imaginary) roots

Type: Complex numbers with real and imaginary parts

Solutions have the form:

a + bi

and

a - bi

Case 3: Δ > 0 and is a Perfect Square

Condition:

b^2 - 4ac = k^2

(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:

b^2 - 4ac > 0

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: $x^2 - 6x + 9 = 0$

Step 1: Identify a = 1, b = -6, c = 9

Step 2: Calculate Δ =

(-6)^2 - 4(1)(9) = 36 - 36 = 0

Step 3: Since Δ = 0, we have one repeated real root

Solution:

x = \frac{6}{2(1)} = 3

(repeated root)

Example 2: $x^2 + 2x + 5 = 0$

Step 1: Identify a = 1, b = 2, c = 5

Step 2: Calculate Δ =

(2)^2 - 4(1)(5) = 4 - 20 = -16

Step 3: Since Δ < 0, we have two complex roots

Solutions:

x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i

Example 3: $x^2 - 5x + 6 = 0$

Step 1: Identify a = 1, b = -5, c = 6

Step 2: Calculate Δ =

(-5)^2 - 4(1)(6) = 25 - 24 = 1

Step 3: Since Δ = 1 (perfect square), we have two rational roots

Solutions:

x = \frac{5 \pm 1}{2}

, so

x = 3

x = 2

Example 4: $x^2 - 4x + 1 = 0$

Step 1: Identify a = 1, b = -4, c = 1

Step 2: Calculate Δ =

(-4)^2 - 4(1)(1) = 16 - 4 = 12

Step 3: Since Δ = 12 (not a perfect square), we have two irrational roots

Solutions:

x = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}

🧠 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!