If a is 0, the equation isn't quadratic — it's linear (bx + c = 0) — so the calculator asks for a non-zero value of a rather than dividing by zero.

What does a negative discriminant mean?

It means the equation has no real solutions — the parabola never crosses the x-axis. The two solutions are complex numbers, shown in the form a ± bi.

What is the discriminant, exactly?

It's the expression b² − 4ac from inside the square root of the quadratic formula. Its sign (positive, zero, or negative) determines whether the equation has two real roots, one repeated real root, or two complex roots.

Can the roots be irrational?

Yes — when the discriminant is positive but not a perfect square, the roots will be real but irrational, shown as a decimal approximation.

How do I verify the answer manually?

Substitute each calculated root back into the original equation ax² + bx + c — the result should equal (or very nearly equal, accounting for rounding) zero for a correct solution.

Math & Statistics Cluster

Quadratic Equation Solver

Solve ax²+bx+c=0 for real or complex roots.

TL;DR: Quadratic Equation Solver is a free, browser-based tool that lets you solve ax²+bx+c=0 for real or complex roots.

Solves equations in the form a×² + b× + c = 0

Understanding the Quadratic Equation Solver

The Quadratic Equation Solver solves mathematical equations, statistical distributions, and numerical relationships with step-by-step arithmetic breakdowns and high-precision calculations.

How It Works

Enter your known values, dataset numbers, or mathematical coefficients.
Select your preferred precision decimal places or calculation mode.
Click to compute and review the step-by-step derivation steps.
Copy the solved numerical answer or explore related mathematical proofs.

Mathematical Formula & Methodology

Here is the standard mathematical equation used to calculate this result accurately:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Formula Variables Explained:
Solves second-degree polynomial equations $ax^2 + bx + c = 0$ with discriminant analysis ( $\Delta = b^2 - 4ac$ ) for real and complex roots.

Step-by-Step Worked Example

For $2x^2 + 5x - 3 = 0$: Discriminant = $25 - 4(2)(-3) = 49$. Roots are $x = \frac{-5 \pm 7}{4} \implies x_1 = 0.5, \, x_2 = -3$ .

How to Get the Best Results

Sample vs Population Data: In statistical variance and standard deviation, use sample formulas (Bessel's correction $n-1$) when analyzing subset samples.
Order of Operations: Computations strictly adhere to standard mathematical PEMDAS / BODMAS operational precedence.
High Precision Arithmetic: Internal calculations use 64-bit IEEE floating-point arithmetic for extreme numerical accuracy.

Related Guides & Tutorials

Percentage Calculator Guide — Learn in-depth concepts, best practices, and expert tips.

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All processing runs locally inside your browser using JavaScript and HTML5 APIs. Your data, files, and inputs are never uploaded to any remote server. Complete privacy by design.

Frequently Asked Questions

: If a is 0, the equation isn't quadratic — it's linear (bx + c = 0) — so the calculator asks for a non-zero value of a rather than dividing by zero.
: It means the equation has no real solutions — the parabola never crosses the x-axis. The two solutions are complex numbers, shown in the form a ± bi.
: It's the expression b² − 4ac from inside the square root of the quadratic formula. Its sign (positive, zero, or negative) determines whether the equation has two real roots, one repeated real root, or two complex roots.
: Yes — when the discriminant is positive but not a perfect square, the roots will be real but irrational, shown as a decimal approximation.
: Substitute each calculated root back into the original equation ax² + bx + c — the result should equal (or very nearly equal, accounting for rounding) zero for a correct solution.

Cite This Tool

Referencing this tool in a paper, article, or bibliography? Copy a ready-made citation below.

NexLove.org. (2026). Quadratic Equation Solver [Software]. https://nexlove.org/tools/quadratic-equation-solver.html

“Quadratic Equation Solver.” NexLove.org, 2026, nexlove.org/tools/quadratic-equation-solver.html.

NexLove.org (2026) Quadratic Equation Solver. Available at: https://nexlove.org/tools/quadratic-equation-solver.html (Accessed: 2026).

@misc{nexlove_quadratic_equation_solver,
  title = {Quadratic Equation Solver},
  author = {{NexLove.org}},
  year = {2026},
  url = {https://nexlove.org/tools/quadratic-equation-solver.html}
}

Embed This Tool

Add the live Quadratic Equation Solver tool to your own website with this snippet — it loads the real, working tool in an iframe, not a static screenshot.

<iframe src="https://nexlove.org/embed/quadratic-equation-solver.html" width="100%" height="600" style="border:1px solid #e2e8f0;border-radius:12px" title="Quadratic Equation Solver — NexLove.org" loading="lazy"></iframe>

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