🔍 Discriminant Calculator
Calculate discriminant of quadratic equation
ax² + bx + c
How to Use This Calculator
Enter Coefficient a
Type the coefficient of x² (a). For example, if your equation is 2x² + 5x + 3, enter 2.
Enter Coefficient b
Type the coefficient of x (b). For 2x² + 5x + 3, enter 5. This can be zero if there's no x term.
Enter Constant c
Type the constant term (c). For 2x² + 5x + 3, enter 3. This can be zero if there's no constant.
Click Calculate Discriminant
Press the button to calculate Δ = b² - 4ac and see the nature of the roots.
Interpret Results
See the discriminant value and whether the equation has two real roots, one real root, or two complex roots.
Formula
Discriminant: Δ = b² - 4ac
Nature of Roots:
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root
- Δ < 0: Two complex conjugate roots
Example 1: x² - 5x + 6 = 0
Given: a = 1, b = -5, c = 6
Δ = (-5)² - 4(1)(6)
Δ = 25 - 24 = 1
Δ > 0 → Two distinct real roots (x = 2, x = 3)
Example 2: x² - 4x + 4 = 0
Given: a = 1, b = -4, c = 4
Δ = (-4)² - 4(1)(4)
Δ = 16 - 16 = 0
Δ = 0 → One repeated real root (x = 2)
Example 3: x² + x + 1 = 0
Given: a = 1, b = 1, c = 1
Δ = (1)² - 4(1)(1)
Δ = 1 - 4 = -3
Δ < 0 → Two complex conjugate roots
About Discriminant Calculator
The Discriminant Calculator computes the discriminant (Δ = b² - 4ac) of a quadratic equation ax² + bx + c = 0. The discriminant reveals crucial information about the nature and number of solutions without solving the equation completely.
When to Use This Calculator
- Quick Analysis: Determine solution type without solving
- Algebra Homework: Check nature of roots before calculating
- Problem Verification: Verify if solutions are real or complex
- Graph Understanding: Understand how parabola relates to x-axis
- Exam Preparation: Quickly analyze quadratic equations
- Mathematical Research: Study properties of quadratic equations
Why Use Our Calculator?
- ✅ Instant Analysis: Know solution type immediately
- ✅ Color-Coded Results: Visual indicators for discriminant types
- ✅ Shows Nature: Clearly states two real, one real, or complex roots
- ✅ Step-by-Step: Displays the discriminant calculation
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding the Discriminant
The discriminant appears inside the square root of the quadratic formula. Its value determines the type and number of solutions:
- Positive Discriminant (Δ > 0): Square root of a positive number gives two different real numbers → Two distinct real roots. The parabola crosses the x-axis at two points.
- Zero Discriminant (Δ = 0): Square root of zero is zero → One repeated real root. The parabola touches the x-axis at exactly one point (the vertex).
- Negative Discriminant (Δ < 0): Square root of a negative number requires imaginary numbers → Two complex conjugate roots. The parabola doesn't cross the x-axis.
Real-World Applications
Physics: In projectile motion, a positive discriminant means the object hits the ground twice (not physically possible), zero means it just touches the ground, and negative means it never reaches ground level.
Engineering: When designing systems, the discriminant helps determine if solutions exist for optimization problems.
Economics: Profit functions use discriminant analysis to determine if break-even points exist.
Tips for Using This Calculator
- The discriminant is the expression under the square root in the quadratic formula
- No need to solve the equation to find the discriminant
- Check the discriminant before attempting to solve
- A negative discriminant means you'll need complex numbers
- The discriminant tells you about the graph's intersection with the x-axis
- Use this to verify if your solutions should be real or complex
Frequently Asked Questions
Why is the discriminant important?
The discriminant tells you the nature of solutions without solving the equation. It shows if roots are real or complex, and if real, whether they're distinct or repeated.
What does a zero discriminant mean?
A zero discriminant (Δ = 0) means the quadratic has exactly one real solution (a repeated root). The parabola touches the x-axis at its vertex but doesn't cross it.
Can the discriminant be negative?
Yes! A negative discriminant means the quadratic equation has no real solutions—only complex conjugate solutions. The parabola doesn't intersect the x-axis.
Do I need to solve the equation to find the discriminant?
No! The discriminant only requires the coefficients a, b, and c. You can determine the nature of roots without actually finding them.
How is the discriminant related to the graph?
The discriminant determines how the parabola relates to the x-axis: crosses twice (Δ > 0), touches once (Δ = 0), or never crosses (Δ < 0).
What if a = 0?
If a = 0, it's not a quadratic equation (it's linear). The discriminant formula only applies to quadratic equations where a ≠ 0.
Can I use the discriminant to find the actual solutions?
The discriminant tells you the type of solutions, but to find the actual roots, use the quadratic formula: x = (-b ± √Δ) / (2a).