Nature of Roots of Quadratic Equations
Explore how the discriminant determines whether roots are real and distinct, real and equal, or complex.
Ch. 4 Quadratic Equations 4.4 Discriminant Analysis Medium
Instructions
- Identify the coefficients a, b, and c from the quadratic equation ax² + bx + c = 0.
- Calculate the discriminant: D = b² - 4ac.
-
Analyze the discriminant to determine the nature of roots:
- If D > 0: Two distinct real roots
- If D = 0: Two equal real roots (repeated root)
- If D < 0: No real roots (complex roots)
- Use the quadratic formula to find the actual roots (if real).
- Observe how the parabola intersects the x-axis based on the discriminant.
Progress
0 of 5 steps completed
Difficulty Level
Why This Matters
Understanding the discriminant is crucial for:
- • Predicting root behavior without solving
- • Graphing quadratic functions accurately
- • Engineering and physics applications
- • Advanced algebra and calculus
Problem
Generating problem...
1
Step 1 — Identify coefficients a, b, c
From the equation ax² + bx + c = 0, identify the values of a, b, and
c.
Hint: Look at each term in the equation:
- The coefficient of x² is 'a'
- The coefficient of x is 'b'
- The constant term is 'c'
- Watch for signs (+ or -) and missing terms
a = b = c =
2
Step 2 — Calculate the discriminant
Calculate D = b² - 4ac step by step
Hint: Break down the discriminant calculation:
- First calculate b² (b squared)
- Then calculate 4ac (4 × a × c)
- Finally, D = b² - 4ac
- Pay attention to positive and negative signs
b² = 4ac =
D = b² - 4ac =
3
Step 3 — Determine nature of roots
Based on the discriminant value, what is the nature of the roots?
Hint: Use these rules:
- If D > 0: Two different real roots (parabola crosses x-axis twice)
- If D = 0: One repeated real root (parabola touches x-axis once)
- If D < 0: No real roots (parabola doesn't touch x-axis)
4
Step 4 — Calculate the roots (if real)
Use the quadratic formula: x = (-b ± √D) / 2a
Hint: Apply the quadratic formula step by step:
- x₁ = (-b + √D) / (2a)
- x₂ = (-b - √D) / (2a)
- If D < 0, the roots are complex (enter 0 for both)
- Calculate √D first, then substitute values
x₁ = x₂ =
Note: For complex roots, enter 0 for both values, or skip this step.
5
Step 5 — Visual Interpretation
Observe the parabola in the visualization below:
Hint: Connect discriminant to visual behavior:
- Count how many times the parabola crosses the x-axis
- Notice the vertex position relative to the x-axis
- Observe the direction the parabola opens (up or down)
- D > 0: Parabola crosses x-axis at two points (two real roots)
- D = 0: Parabola touches x-axis at exactly one point (vertex on x-axis)
- D < 0: Parabola does not touch x-axis (no real roots)
Visualization (GeoGebra)
⚠️ GeoGebra visualization unavailable
The interactive graph cannot be loaded. You can still complete the steps above.
Discriminant Quick Reference
D > 0
Two distinct real roots
Parabola crosses x-axis twice
D = 0
Two equal real roots
Parabola touches x-axis once
D < 0
No real roots
Parabola doesn't touch x-axis
Concept Mastery
Coefficient Identification 0 correct attempts
Discriminant Calculation 0 correct attempts
Nature of Roots Analysis 0 correct attempts
Root Calculation 0 correct attempts
Visual Interpretation 0 problems completed
Overall Mastery: 0%