quadratic equation calculator online

What this quadratic equation calculator does

This online quadratic solver helps you quickly solve equations in standard form: ax² + bx + c = 0. Instead of manually calculating each step, you can enter your coefficients and immediately see:

• The equation you entered
• The discriminant value D = b² - 4ac
• The nature of roots (two real, one real repeated, or complex)
• The actual root values

Quadratic formula used by the calculator

The tool uses the classic quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

The expression under the square root, b² - 4ac, is called the discriminant. It tells us what kind of roots to expect.

How to interpret the discriminant

• D > 0: two distinct real roots
• D = 0: one repeated real root
• D < 0: two complex conjugate roots

How to use this calculator online

1. Type values for a, b, and c.
2. Click Calculate Roots.
3. Read the discriminant and roots in the result box.
4. Use Reset Example to load a sample equation.

Tip: Press Enter while focused on any input field to calculate instantly.

Worked examples

Example 1: x² - 3x + 2 = 0

Here, a = 1, b = -3, c = 2. The discriminant is D = (-3)² - 4(1)(2) = 9 - 8 = 1. Since D > 0, there are two real roots: x = 1 and x = 2.

Example 2: 2x² + 4x + 2 = 0

Here, a = 2, b = 4, c = 2. D = 4² - 4(2)(2) = 16 - 16 = 0. That means one repeated root: x = -1.

Example 3: x² + 2x + 5 = 0

Here, a = 1, b = 2, c = 5. D = 2² - 4(1)(5) = 4 - 20 = -16. Since D is negative, roots are complex: x = -1 ± 2i.

Common mistakes when solving quadratic equations

• Forgetting that a cannot be 0 for a true quadratic equation
• Sign errors in -b or in b² - 4ac
• Forgetting parentheses around negative coefficients
• Rounding too early and losing precision

This calculator minimizes sign mistakes and gives clean, readable output for both real and complex roots.

Why use an online quadratic calculator?

A fast web-based solver is useful for homework checks, exam preparation, teaching demos, and engineering calculations. You can verify answers in seconds and focus on understanding the method rather than arithmetic slips.

Frequently asked questions

Can this solve linear equations too?
Yes. If you enter a = 0, the tool attempts to solve bx + c = 0 and explains special cases.

Does it support decimals and negative numbers?
Absolutely. You can use integers, decimals, and negative coefficients.

Does it show complex roots?
Yes. For negative discriminant values, the output shows roots in the form p ± qi.

Final thoughts

If you need a reliable quadratic equation calculator online, this tool gives instant, accurate results with clear interpretation. Enter coefficients, click calculate, and get discriminant and roots immediately.