solve quadratic formula calculator

What this solve quadratic formula calculator does

This tool helps you solve any equation in the standard form ax² + bx + c = 0. A quadratic equation appears in algebra, physics, finance, engineering, and data modeling. Instead of manually calculating square roots and signs every time, this calculator gives you roots instantly and shows the key values used in the process.

It supports the three main outcomes you expect from a quadratic solver:

• Two distinct real roots when the discriminant is positive.
• One repeated real root when the discriminant is exactly zero.
• Two complex roots when the discriminant is negative.

The quadratic formula explained

The expression under the square root, b² - 4ac, is called the discriminant. It determines the nature of the roots.

Discriminant quick guide

• If D = b² - 4ac > 0: two different real roots.
• If D = 0: one repeated real root (also called a double root).
• If D < 0: complex conjugate roots.

How to use this calculator

1. Enter numeric values for a, b, and c.
2. Click Calculate Roots.
3. Read the displayed discriminant and resulting root values.
4. Use Clear to reset or Load Example for a quick test case.

Tip: If a = 0, the equation is not quadratic. The calculator automatically treats it as a linear equation of the form bx + c = 0.

Worked examples

Example 1: Two real roots

Equation: x² - 5x + 6 = 0
Here, a = 1, b = -5, c = 6.
Discriminant: D = 25 - 24 = 1, so there are two real roots: x = 2 and x = 3.

Example 2: One repeated root

Equation: x² - 4x + 4 = 0
Discriminant: D = 16 - 16 = 0.
The root is repeated: x = 2.

Example 3: Complex roots

Equation: x² + 2x + 5 = 0
Discriminant: D = 4 - 20 = -16.
Since D is negative, roots are complex: x = -1 ± 2i.

Common mistakes to avoid

• Forgetting to include a coefficient as zero (for example, if there is no x-term).
• Mixing up signs in -b and ±.
• Not using parentheses around the full numerator and denominator.
• Treating a linear equation as quadratic when a = 0.

Why this solver is useful for students and professionals

Whether you are checking homework, validating spreadsheet models, or building equations from measured data, quick and accurate root calculation matters. This solve quadratic formula calculator reduces arithmetic errors and makes interpretation easier by clearly showing the discriminant and solution type.

FAQ

Can I enter decimals or negative numbers?

Yes. The inputs accept integers and decimals, including negative values.

Does this calculator show complex roots?

Yes. When the discriminant is negative, it returns roots in the form real ± imaginary i.

Is it suitable for checking exam practice?

Yes. It is great for verification after solving by hand. For learning, try solving first manually, then use this page to confirm your answers.