Quadratic Equation Calculator 2025: Complete Guide
The quadratic equation ax² + bx + c = 0 is central to algebra. Our Quadratic Equation Calculator solves it instantly using the quadratic formula, completing the square, or factoring — with full step-by-step explanations, vertex, discriminant, and graph insights.
The Quadratic Formula
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Where a ≠ 0. Works for all real and complex roots.
Discriminant: Nature of Roots
D = b² - 4ac
- D > 0: Two distinct real roots
- D = 0: One real root (repeated)
- D < 0: Two complex roots
Vertex of Parabola
x = -b/(2a)
y = f(x) at vertex
Step-by-Step: Solve x² - 5x + 6 = 0
- a=1, b=-5, c=6
- D = (-5)² - 4(1)(6) = 25 - 24 = 1 > 0
- x = [5 ± √1]/2 → x = 3, x = 2
- Roots: x = 2, 3
Completing the Square
Rewrite ax² + bx + c = 0 as a(x - h)² + k = 0.
Example: x² + 6x + 5 = 0
→ x² + 6x = -5
→ (x + 3)² - 9 = -5
→ (x + 3)² = 4
→ x + 3 = ±2 → x = -1, -5
Factoring Method
Find two numbers that multiply to ac and add to b.
Example: 2x² + 7x + 3 = 0
→ AC = 6 → 1×6, 2×3
→ 2x² + 6x + x + 3 = 0
→ 2x(x + 3) + 1(x + 3) = 0
→ (2x + 1)(x + 3) = 0 → x = -1/2, -3
Real-World Applications
- Physics: Projectile motion (height = -4.9t² + v₀t + h₀)
- Engineering: Beam deflection
- Finance: Break-even analysis
- Optimization: Max profit/min cost
Common Mistakes
- Forgetting
±in formula - Wrong sign in discriminant
- Dividing by zero (a=0)
- Not simplifying radicals
Advanced Tools
FAQs
Can it handle complex roots? Yes — e.g., x² + 1 = 0 → x = ±i
What if a = 0? Becomes linear: solve bx + c = 0
Can I enter fractions? Yes — use decimals or fractions (1/2).
Conclusion
Mastering quadratics is key to algebra success. Our 2025 Quadratic Equation Calculator delivers instant, accurate, step-by-step solutions for any quadratic. From homework to real-world modeling, solve with confidence. Enter your coefficients now!
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