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Quadratic Solver.

Solve quadratic equations ax² + bx + c = 0

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Solve: ax² + bx + c = 0

1x² -5x + 6 = 0

Discriminant Analysis

Discriminant (Δ)1.0000
Nature of Roots✓ Two Distinct Real Roots

Solution x₁

3.0000

Solution x₂

2.0000

Parabola Properties

Vertex(2.5000, -0.2500)
Axis of Symmetryx = 2.5000
Y-intercept(0, 6)
Opens↑ Upward (Minimum)

Understanding Quadratic Equations

Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a. This formula solves any quadratic equation ax² + bx + c = 0.
Discriminant (Δ): The value Δ = b² - 4ac determines the nature of roots:
  • • Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
  • • Δ = 0: One repeated real root (parabola touches x-axis once)
  • • Δ < 0: Two complex roots (parabola doesn't touch x-axis)
Vertex Form: The vertex represents the minimum (a > 0) or maximum (a < 0) point of the parabola. It's located at x = -b/(2a).

Use Cases

Homework and Exam Preparation

Quickly verify your manual solutions to quadratic equations, check your work, and understand the step-by-step process.

Example: Solve x² - 5x + 6 = 0 to get roots 2 and 3.

Engineering and Physics Calculations

Find roots of quadratic equations that arise in projectile motion, circuit analysis, and optimization problems.

Example: Determine the time when a projectile hits the ground using h = -4.9t² + 20t + 1.

Frequently Asked Questions

How do I use the Quadratic Solver?
Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The solver will compute the roots using the quadratic formula and display the solutions, including complex roots if the discriminant is negative.
What if the discriminant is negative?
If the discriminant (b² - 4ac) is negative, the equation has no real roots. The solver will show the complex roots in the form of a + bi, where i is the imaginary unit.
Can the solver handle a = 0?
If a = 0, the equation is not quadratic but linear. The solver will indicate that the equation is not quadratic and may not provide a solution. Ensure a ≠ 0 for a valid quadratic equation.

Tips & Common Mistakes

Tips

  • Ensure the equation is in standard form ax² + bx + c = 0 before entering coefficients.
  • Double-check the signs of b and c; a common error is entering a negative b as positive.
  • Use the discriminant (b² - 4ac) to predict the nature of roots: positive for two real, zero for one real, negative for complex.
  • If a = 0, the equation is linear, not quadratic, so the solver may not work correctly.

Common Mistakes to Avoid

  • Entering coefficients in the wrong order (e.g., swapping a and c).
  • Forgetting to include the sign of the coefficient (e.g., entering 5 instead of -5 for b).
  • Assuming the solver can handle a = 0; it's designed for quadratic equations only.

Last updated: August 13, 2026