Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0. Get discriminant, root type, and exact numerical roots — including complex conjugates.

Quadratic Equation Calculator

Enter coefficients a, b, c for ax² + bx + c = 0

Coefficient a

Note: a ≠ 0

Coefficient b

Coefficient c

Please enter numeric values for a, b, and c.

Key Features

Compute discriminant (Δ) and classify root types

Handles all cases: two real roots, double root, complex roots

Copy results to clipboard for assignments and reports

Example presets for Δ > 0, Δ = 0, Δ < 0

Understanding Quadratics

Standard Form

A quadratic equation has the standard form ax² + bx + c = 0 where a ≠ 0.

Root Types

Δ > 0: Two distinct real roots
Δ = 0: One real double root
Δ < 0: A pair of complex conjugate roots

Common Mistakes

Using a = 0 (not quadratic)
Sign errors when computing b² − 4ac
Forgetting parentheses in −b/(2a)
Confusing √(4ac − b²) with √(b² − 4ac) in complex form

Tips & Best Practices

Use example presets to validate the three discriminant cases quickly.

Keep parentheses around −b and 2a when evaluating the formula.

Interpret Δ < 0 roots as complex numbers p ± qi.

Common Use Cases

Algebra homework checks: verify exams and practice problems

STEM education: demonstrate real/complex roots and discriminant behavior

Engineering prelims: quick checks for quadratic characteristic equations

How to Use the Quadratic Equation Solver

Step-by-Step Instructions

1Enter coefficients a, b, and c into the input fields
2Confirm a ≠ 0; otherwise adjust the equation
3Review Δ and root results; try presets to test scenarios
4Copy results for reports or classwork

Pro Tips

Use exact fractions for symbolic math; this tool provides numeric evaluation to six decimals
For Δ < 0, interpret roots as p ± qi where p = −b/(2a) and q = √(4ac − b²)/(2a)

Step-by-Step Instructions

Enter Coefficient A

Input the coefficient of x² (the number before x²). This cannot be zero, as it wouldn't be a quadratic equation.

Enter Coefficient B

Input the coefficient of x (the number before x). This can be positive, negative, or zero.

Enter Constant C

Input the constant term (the number without x). This can also be positive, negative, or zero.

Solve the Equation

Click "Solve Quadratic Equation" to calculate the roots using the quadratic formula.

Interpret Results

Review the discriminant and roots. The discriminant tells you if roots are real, complex, or repeated.

Check the Graph

Examine the parabola visualization to understand where the equation crosses the x-axis (the roots).

Pro Tips

🔢

Standard Form: Make sure your equation is in the form ax² + bx + c = 0 before entering coefficients.

📊

Discriminant Analysis: Positive discriminant = 2 real roots, zero = 1 repeated root, negative = 2 complex roots.

🎯

Verify Your Answers: Substitute the roots back into the original equation to confirm they equal zero.

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Visual Learning: Use the graph to see how the parabola's shape relates to the number and type of roots.

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Complex Numbers: Don't worry if you get complex roots - they're still valid mathematical solutions.

Frequently Asked Questions

What does the discriminant tell me?

The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 for two real roots, Δ = 0 for one real double root, and Δ < 0 for two complex conjugate roots.

Can the solver handle negative and decimal coefficients?

Yes. You can enter any real numbers for a, b, and c. The calculator computes stable numeric results to six decimal places.

Why must a ≠ 0?

When a = 0, the equation becomes linear (bx + c = 0), not quadratic. This tool specifically solves quadratic equations where a ≠ 0.

Do you show steps?

We show Δ computation and substitution into the quadratic formula to illustrate the process behind the final roots.

How do you handle very large or tiny coefficients?

Results are rounded to six decimals for readability. Extremely large or small values render in standard numeric form; for exact algebra, consider symbolic methods while using this tool for numeric verification.