Quadratic Formula Calculator
Enter the coefficients a, b, and c to solve a quadratic equation and see its roots, discriminant, vertex, and calculation steps.
Two distinct real roots
- Discriminant: (−5)² − 4(1)(6) = 1.
- x = (5 ± √1) / 2.
- x₁ = 2 and x₂ = 3.
Embed Quadratic Calculator
Copy this iframe code into a page that permits embeds.
All calculations run locally in your browser. Your entered values are not sent to an external service.
How to Use the Quadratic Formula Calculator
Write the equation in standard form, ax² + bx + c = 0, and enter the three coefficients. Include negative signs. If a term is missing, enter zero for its coefficient. For x² − 9 = 0, enter a = 1, b = 0, and c = −9.
- Move every term to one side so the other side equals zero.
- Identify a as the coefficient of x², b as the coefficient of x, and c as the constant.
- Enter the coefficients and select Solve Quadratic.
- Review the discriminant, roots, vertex, and substitution steps.
Coefficient a cannot be zero. If a equals zero, the equation is linear rather than quadratic. Use the equation solver for that case.
Quadratic Formula and Standard Form
The plus-or-minus sign produces two values when the square-root term is nonzero. One calculation adds the square root and the other subtracts it. When the discriminant is zero, both calculations produce the same repeated root.
The formula works for every quadratic equation with real coefficients, including equations that do not factor neatly. Factoring may be faster for simple integer roots, while completing the square helps explain where the formula comes from.
How the Discriminant Predicts the Roots
The discriminant is D = b² − 4ac. Its sign predicts the number and type of solutions before the complete formula is evaluated.
| Discriminant | Root type | Graph meaning |
|---|---|---|
| D > 0 | Two distinct real roots | The parabola crosses the x-axis twice. |
| D = 0 | One repeated real root | The vertex touches the x-axis. |
| D < 0 | Two complex conjugate roots | The parabola has no real x-intercepts. |
A positive perfect-square discriminant gives rational roots when the coefficients are rational. A positive non-square discriminant usually produces irrational roots.
Quadratic Formula Examples
Two real roots
For x² − 5x + 6 = 0, a = 1, b = −5, and c = 6. The discriminant is 25 − 24 = 1. The formula gives x = (5 ± 1) / 2, so the roots are 2 and 3.
One repeated root
For x² + 2x + 1 = 0, the discriminant is 4 − 4 = 0. The formula gives x = −2 / 2 = −1. This root occurs twice, and the parabola touches the x-axis at its vertex.
Two complex roots
For x² + 1 = 0, the discriminant is −4. Since √(−4) = 2i, the roots are x = i and x = −i. These roots are complex conjugates.
Decimal coefficients
For 0.5x² + x − 1.5 = 0, the discriminant is 4. The formula produces roots −3 and 1. Decimals are valid as long as a is not zero.
Vertex and Axis of Symmetry
The calculator also reports the vertex. Its x-coordinate is xᵥ = −b / (2a), which is also the axis of symmetry. Substitute that value into ax² + bx + c to find the y-coordinate. If a is positive, the parabola opens upward and the vertex is a minimum. If a is negative, it opens downward and the vertex is a maximum.
The vertex does not replace the roots. Roots describe where the graph meets the x-axis, while the vertex describes the turning point.
When Quadratic Equations Are Useful
Quadratic equations appear whenever a quantity changes at a rate that is itself changing. In physics, a height model such as h(t) = −16t² + v₀t + h₀ can describe a projectile under constant gravity. Solving h(t) = 0 estimates when the object reaches the ground. The physically meaningful answer must also satisfy the time restrictions of the problem.
In geometry, a fixed-area or perimeter condition can create a quadratic equation for an unknown side length. Revenue and profit models may also be quadratic when price and demand change together. In each application, the roots are mathematical candidates. Units, domain restrictions, and the original situation determine which candidates are usable.
Exact values and decimal values
The formula may produce exact radical forms such as (3 + √5) / 2. Decimal results are convenient for measurement, but they are approximations when the radical is irrational. Keep an exact form during algebra work and round only at the end using the precision required by the problem.
Checking the solutions
Substitute each reported root into the original equation. A valid root makes the left side zero, apart from small rounding differences for decimal approximations. Checking is especially important after moving terms into standard form because a copied sign can change both the discriminant and the roots.
Choosing a solving method
Use factoring when the expression has obvious integer factors. Use the square-root property for equations such as (x − 3)² = 7. Completing the square is helpful for rewriting a quadratic in vertex form. The quadratic formula is the dependable general method because it works whether the roots are rational, irrational, repeated, or complex.
Common Quadratic Formula Mistakes
Using the wrong sign for b
Copy b with its sign, then compute −b. If b is −5, then −b is positive 5.
Forgetting that the denominator is 2a
The entire numerator is divided by 2a. Do not divide only the square-root term.
Calculating b² incorrectly
Square the signed value. For b = −5, b² = (−5)² = 25, not −25.
Leaving the equation outside standard form
Move all terms to one side first. The calculator expects coefficients for an equation equal to zero.
Treating a negative discriminant as an error
A negative discriminant means the roots are complex, not that the equation is invalid.
Quadratic Formula Calculator FAQs
What does this calculator solve?
It solves equations in the form ax² + bx + c = 0 with real numerical coefficients and a not equal to zero.
What if a equals zero?
The equation becomes linear, so use a linear equation solver instead.
What is the discriminant?
The discriminant is b² − 4ac. It predicts whether the roots are two real values, one repeated value, or two complex values.
Can the calculator show complex roots?
Yes. A negative discriminant is reported as a conjugate pair in the form p ± qi.
Can I use decimal coefficients?
Yes. Enter any finite decimal values, provided a is not zero.
Why can one root appear twice?
When the discriminant is zero, the plus and minus versions of the formula are identical, producing a repeated root.
Are calculations sent to a server?
No. The calculation runs locally in the browser.
Related Algebra Calculators
Slope CalculatorRise, run, angle, and line equation.
System of Equations SolverTwo linear equations in x and y.
Factoring CalculatorFactor ax² + bx + c over the integers.
Midpoint CalculatorMidpoint of a segment.
Simplify CalculatorExpand and combine like terms.
Explore More Algebra Tools
Continue with equation, factoring, exponent, and coordinate tools in the SolverCalc algebra collection.
Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.