Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Finds both real and complex roots with step-by-step solutions.
Explore roots and the parabola
ax² + bx + c = 0
Roots: 2, 3
Nature of the roots: Two distinct real roots
Discriminant: 1. Vertex: (2.5, -0.25).
Real roots are the x-intercepts. A repeated root touches the axis at the vertex.
Graph coordinates
| x | y |
|---|---|
| -0.5 | 8.75 |
| 0.25 | 4.8125 |
| 1 | 2 |
| 1.75 | 0.3125 |
| 2.5 | -0.25 |
| 3.25 | 0.3125 |
| 4 | 2 |
| 4.75 | 4.8125 |
| 5.5 | 8.75 |
Shared inputs are stored in the link fragment. Anyone receiving the link can read them.
Learn with a worked scenario
Real or complex roots of ax² + bx + c = 0, with a real-coordinate parabola preview.
Question
Solve x² − 6x + 5 = 0.
- Discriminant = (−6)² − 4 × 1 × 5 = 16.
- x = (6 ± 4) / 2, giving 1 and 5.
- Vertex x = 3; y = 3² − 6 × 3 + 5 = −4.
Roots 1 and 5; vertex (3, −4). Both real roots are x-intercepts.
Precision: Roots, the discriminant and the vertex are shown to at most 8 significant digits; the calculation itself is not rounded.
Load this example in the calculatorTry a variation
Keep a = 1 and b = −6, but set c = 10. Where does the vertex move, and what are the roots? Predict the answer before you reveal it.
Predict, then reveal
The vertex moves up to (3, 1), above the x-axis, so the parabola has no x-intercepts. The discriminant is 36 − 40 = −4, giving the complex roots 3 ± i.
| c, with a = 1 and b = −6 | Roots |
|---|---|
| 5 | 1 and 5 |
| 9 | 3 (repeated) |
| 10 | 3 ± i |
Check the assumptions
- a must be nonzero.
- The graph shows real x and y, not complex roots.
- The plot is a local preview; consult its coordinate range and table.
Common mistake: Using the absolute value of a negative discriminant and reporting real roots.
Method reference: OpenStax College Algebra: Quadratic Functions. These examples illustrate the method; they do not validate every possible input.
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Quadratic Formula Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.
Review your inputs
Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Quadratic Formula Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Variables: a = coefficient of x², b = coefficient of x, c = constant term
Discriminant
Δ = b² - 4ac
Variables: Determines the nature and number of roots
When to Use This Calculator
- •Use the Quadratic Formula Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
Solve a nondegenerate quadratic for real or complex roots, then compare its discriminant, vertex and sampled real-coordinate graph. The coordinate table provides an alternative to the plot.
The Theory Behind It
For ax²+bx+c=0 with a nonzero, the discriminant D=b²−4ac determines the root type. Positive D gives two real roots, zero gives a repeated root, and negative D gives a conjugate pair of complex roots. The vertex has x-coordinate −b/(2a); substituting that x into the polynomial gives its y-coordinate. The implementation uses a numerically stable form for real roots to reduce cancellation, and reports the real and imaginary parts separately for complex roots. A complex root is not an x-intercept of the real-coordinate parabola. The graph is a sampled local view centered on the vertex, with a coordinate table; its horizontal and vertical scales may differ. Very large coordinate ranges can suppress the graph while preserving supported numerical results. Display rounding is separate from the internal calculation. Set a to a nonzero value; linear equations require a different model. a must be nonzero. The graph shows real x and y, not complex roots. The plot is a local preview; consult its coordinate range and table.
Real-World Applications
- •Real or complex roots of ax² + bx + c = 0, with a real-coordinate parabola preview.
Frequently Asked Questions
What does a negative discriminant mean?
There are two complex conjugate roots and no real x-intercepts. The real parabola still has a vertex and can be graphed.
Why does the graph use a local range?
The view is centered on the vertex to show the quadratic's shape and nearby intercepts. Consult the coordinate table and stated range rather than assuming equal axis scales.
Worked Examples
Example 1: Work through the linked example
Solve x² − 6x + 5 = 0.
Roots 1 and 5; vertex (3, −4). Both real roots are x-intercepts.
Common Mistakes & Tips
- !Using the absolute value of a negative discriminant and reporting real roots.
- !a must be nonzero.
- !The graph shows real x and y, not complex roots.
- !The plot is a local preview; consult its coordinate range and table.
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