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Quadratic Formula Calculator

Enter the coefficients for ax^2 + bx + c = 0. For a true quadratic, a must not be 0. If a = 0, the equation reduces to a linear form.

This calculator supports real and complex roots. Precision affects output rounding only; calculations use full floating-point precision internally.

Use our Quadratic Formula Calculator to quickly solve equations of the form ax^2 + bx + c = 0. Whether you are checking homework, validating engineering computations, or exploring algebra, this tool returns roots, discriminant, and clear interpretation in seconds.

What Is the Quadratic Formula?

The quadratic formula is a universal method for solving any quadratic equation of the form ax^2 + bx + c = 0 with a \u2260 0. The solutions (also called roots) are obtained by substituting the coefficients a, b, and c into the formula: x = (\u2212b \u00b1 \u221a(b^2 \u2212 4ac)) / (2a). The expression under the square root, b^2 \u2212 4ac, is called the discriminant and determines the nature of the roots: two distinct real roots when it is positive, one repeated real root when it equals zero, and two complex conjugate roots when it is negative.

How to Use the Quadratic Formula Calculator

  1. Enter the coefficient a (must be nonzero for a quadratic).
  2. Enter the coefficient b.
  3. Enter the constant term c.
  4. Optionally set decimal precision for the displayed results.
  5. Check \"Show calculation steps\" if you want a step-by-step breakdown.
  6. Click \"Calculate Roots\" to view solutions, discriminant, and interpretation.

Understanding Your Results

Discriminant and Nature of Roots

  • Discriminant > 0: Two distinct real roots. The graph of y = ax^2 + bx + c crosses the x-axis at two points.
  • Discriminant = 0: One real root with multiplicity two. The parabola just touches the x-axis (tangent).
  • Discriminant < 0: Two complex conjugate roots. The parabola does not intersect the x-axis.

Precision and Rounding

The precision setting affects how many decimal places are displayed. Internally, the calculator performs full floating-point computations, then rounds the output for readability. For exact symbolic forms, consider complementing this tool with algebraic simplification, but for most practical needs, a rounded decimal is ideal.

Why Use This Calculator?

  • Speed and accuracy for homework checks and exam prep.
  • Handles real and complex solutions automatically.
  • Optional step-by-step display to reinforce learning.
  • Clear interpretation of the discriminant and root types.

Example

Suppose a = 1, b = -3, and c = 2. The discriminant is b^2 - 4ac = 9 - 8 = 1, which is positive. The roots are x = (3 \u00b1 1) / 2, giving x = 1 and x = 2. If you instead try a = 1, b = 2, c = 5, the discriminant is 4 - 20 = -16, so the solutions are complex: x = -1 \u00b1 2i.

Common Applications

  • Physics: projectile motion, kinematics, and optics.
  • Engineering: optimization of parabolic components and control systems.
  • Finance: time value problems that reduce to quadratic equations.
  • Computer graphics: intersection tests and curve analysis.

Tips for Reliable Results

  • Double-check the sign of each coefficient before calculating.
  • Adjust precision to suit the problem context; more is not always better.
  • If a = 0, the equation is linear, and the calculator will solve bx + c = 0.

With a clean interface, instant feedback, and clear explanations, this Quadratic Formula Calculator is a reliable companion for students, educators, and professionals alike.

FAQs

How do I use the Quadratic Formula Calculator for ax^2 + bx + c = 0?

Enter a, b, and c, set precision if needed, optionally show steps, then click Calculate to see roots and discriminant.

Can the Quadratic Formula Calculator find complex roots?

Yes. If the discriminant is negative, it returns complex conjugate roots with the chosen precision.

What does the discriminant shown by the Quadratic Formula Calculator mean?

It indicates the nature of roots: positive gives two real, zero gives one real double root, negative gives complex roots.

Does the Quadratic Formula Calculator handle cases where a = 0?

Yes. It switches to the linear equation bx + c = 0 and solves for x when a = 0.

How accurate are the results from the Quadratic Formula Calculator?

Calculations use full precision internally; the precision setting only rounds the displayed output.

Can I see the steps in the Quadratic Formula Calculator?

Enable the Show calculation steps option to view a concise breakdown of the formula substitution and solution.

What input format should I use in the Quadratic Formula Calculator?

Use numeric values for a, b, and c. Decimals and negative numbers are supported.

Will the Quadratic Formula Calculator sort the roots?

Yes. For two real roots, it orders them from smallest to largest for readability.