Math & Algebra Tool

Quadratic Formula Solver

Solve ax² + bx + c = 0.

x² + x + = 0

Σ The Formula

x = (-b ± √(b² - 4ac)) / 2a

Real World Examples

Standard Form
x² + 5x + 6 = 0 → x = -2 or x = -3
No Real Roots
x² + 2x + 5 = 0 → Complex roots: -1 ± 2i
Perfect Square
x² - 6x + 9 = 0 → x = 3 (double root)
Physics Application
-16t² + 64t + 80 = 0 → t = 5s (projectile motion)

# About This Calculator

The quadratic formula is one of the most important tools in algebra, discovered independently by mathematicians across ancient Babylonia, Greece, India, and the Islamic world. It provides a universal method to find the roots (solutions) of any quadratic equation in the form ax² + bx + c = 0.

Quadratic equations appear everywhere in real-world applications: calculating projectile trajectories in physics, optimizing profit in business, designing parabolic antennas, modeling population growth, and solving countless engineering problems. Understanding how to solve them is fundamental to STEM education.

The formula x = (-b ± √(b² - 4ac)) / 2a always works, regardless of whether roots are real, complex, rational, or irrational. The discriminant (b² - 4ac) tells you what type of solutions to expect: positive means two real roots, zero means one repeated root, negative means two complex roots.

While factoring is faster when possible, the quadratic formula is your reliable backup that never fails. This calculator handles all cases automatically, including complex numbers, and shows you the discriminant to help you understand the nature of the solutions.

How To Use

  1. Enter coefficients a, b, c.
  2. Click Solve.

Frequently Asked Questions

When do I use the quadratic formula vs factoring?+

Try factoring first if you can spot factors quickly (like x² + 5x + 6 = (x+2)(x+3)). Use the quadratic formula when factoring isn't obvious, when coefficients are decimals, or when you need exact irrational roots. The formula always works!

What does the discriminant tell me?+

The discriminant (b² - 4ac) predicts solution types: If positive, you get two different real roots. If zero, one repeated real root (perfect square). If negative, two complex conjugate roots. It's like a preview of what to expect!

Why are there sometimes two solutions?+

Quadratic equations represent parabolas, which can cross the x-axis at 0, 1, or 2 points. Each crossing is a solution. For example, x² = 4 has solutions x = 2 and x = -2 because both 2² and (-2)² equal 4.

How do I know if my quadratic has real solutions?+

Check the discriminant: b² - 4ac. If it's ≥ 0, you have real solutions. If it's negative, solutions are complex (involving i = √-1). Real-world physics problems usually have real solutions; abstract math can have either.

What are imaginary/complex solutions?+

When the discriminant is negative, you get complex numbers like -1 ± 2i. The 'i' represents √-1. While not 'real' numbers, they're crucial in engineering (AC circuits, signal processing) and quantum physics. This calculator handles them automatically.

Can I solve cubic or higher-degree polynomials this way?+

No, the quadratic formula only works for degree-2 polynomials (highest power is x²). Cubic equations (x³) have their own formulas (much more complex), and degree-5+ generally require numerical methods or specialized techniques.

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