Algebraic Equation Solver (Linear & Quadratic)
Find exact roots for linear and quadratic equations with discriminant analysis.
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How to Use the Equation Solver: Linear & Quadratic Roots Calculator
- 1Select equation type (Quadratic ax² + bx + c = 0 or Linear ax + b = 0).
- 2Enter the numerical coefficients a, b, and c.
- 3Read the calculated real or complex conjugate roots x₁ and x₂.
- 4Examine the discriminant value (Δ) to understand root multiplicity and stability.
Formula & How It Works
Quadratic Formula & Discriminant Analysis
Calculates roots algebraically with full support for negative discriminants (complex roots).
Worked Examples
Solving x² - 5x + 6 = 0
Input: a = 1, b = -5, c = 6
Explanation: b² - 4ac = 25 - 24 = 1. Two distinct real roots: (5 ± 1) / 2 = 3 and 2.
Frequently Asked Questions
What happens when coefficient a is zero in a quadratic equation?▼
If a = 0, the equation degenerates into a linear equation bx + c = 0 with a single solution x = -c/b.
What does a negative discriminant mean?▼
A negative discriminant means the equation has no real solutions; its roots exist in the complex plane as a pair of conjugate imaginary numbers.
How do you find the vertex of a parabola from its coefficients?▼
The x-coordinate of the vertex is located at x = -b / (2a).
Can this solve linear equations?▼
Yes, select 'Linear' mode to solve standard first-degree equations of the form ax + b = 0.
Quick Reference & Conversion Table
| Discriminant Δ | Root Nature | Parabola Geometry | Example |
|---|---|---|---|
| Δ > 0 | Two distinct real roots | Crosses x-axis twice | x² - 5x + 6 = 0 |
| Δ = 0 | One repeated real root | Touches x-axis at vertex | x² - 4x + 4 = 0 |
| Δ < 0 | Two complex conjugate roots | Does not intersect x-axis | x² + 2x + 5 = 0 |
⚠️ Common Pitfalls to Avoid
- •Forgetting to rearrange equations into standard form (ax² + bx + c = 0) before identifying coefficients.
- •Mishandling negative signs when evaluating -b and b².
💡 Practical Real-World Uses
- •Calculating projectile motion flight times and peak parabolic apexes in physics.
- •Determining economic break-even points where cost equals revenue curves.
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