Advanced Quadratic Formula Tool
Enter coefficients for ax² + bx + c = 0. The calculator returns roots, discriminant, vertex, axis, intercept, and checks.
Example Data Table
Use these examples to compare real, repeated, and complex root cases.
| Equation | a | b | c | Discriminant | Root Type | Roots |
|---|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 | 1 | Two real roots | 3, 2 |
| x² + 2x + 1 = 0 | 1 | 2 | 1 | 0 | Repeated root | -1 |
| 2x² + 4x + 5 = 0 | 2 | 4 | 5 | -24 | Complex roots | -1 ± 0.707107i |
| -3x² + 6x + 9 = 0 | -3 | 6 | 9 | 144 | Two real roots | -1, 3 |
Formula Used
Standard equation: ax² + bx + c = 0
Discriminant: D = b² - 4ac
Quadratic formula: x = (-b ± √D) / 2a
Vertex: h = -b / 2a, then k = f(h)
Root sum and product: x₁ + x₂ = -b / a, and x₁x₂ = c / a
If D is positive, the equation has two real roots. If D is zero, it has one repeated root. If D is negative, it has two complex roots.
How to Use This Calculator
- Enter the values of a, b, and c from your equation.
- Make sure the equation is written as ax² + bx + c = 0.
- Select the decimal precision you want for rounded answers.
- Choose whether to show exact roots and solution steps.
- Enter a check value if you want to evaluate f(x).
- Press the calculate button to show the result above the form.
- Use CSV or PDF buttons to save the output.
Quadratic Formula Calculator Guide
Understanding Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0. The value of a cannot be zero. The graph is a parabola. It may open upward or downward. Its roots are the x values where the graph meets the x axis. Some equations have two real roots. Some have one repeated root. Others have two complex roots.
Why the Formula Helps
Factoring is useful, but it does not always work quickly. Completing the square is reliable, yet it can feel long. The quadratic formula gives one direct method. It uses a, b, and c only. This makes it ideal for checking homework, testing models, and preparing clean algebra steps. The discriminant, b² - 4ac, tells the root type before roots are written.
What This Tool Reports
This calculator gives exact radical form when possible. It also gives decimal roots with your chosen precision. It shows the discriminant, root nature, vertex, axis of symmetry, y intercept, sum of roots, and product of roots. These values help you understand the equation, not just the final answer. The tool also formats a step by step solution, so you can follow each substitution.
Practical Uses
Quadratics appear in motion, area, revenue, construction layouts, and many school problems. A small sign error can change the answer. Enter each coefficient carefully. Use a negative sign when needed. Review the formatted equation before trusting the output. If a is zero, the expression is linear, so this calculator warns you instead of forcing a false quadratic result.
Better Checking Habits
After solving, substitute a root back into the equation. The result should be close to zero. With decimals, tiny differences can appear because of rounding. Use more precision when roots are close together. Download the CSV when you need table data. Download the PDF when you need a neat report for class, tutoring, or records.
Common Input Tips
Keep units consistent when coefficients come from measurements. Do not round early in multi step work. Rounded coefficients can move roots noticeably. Compare the vertex value with the roots. This shows whether your parabola crosses, touches, or misses the x axis. Save examples to build practice sets later.
FAQs
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves equations written as ax² + bx + c = 0, where a is not zero.
What does the discriminant mean?
The discriminant is b² - 4ac. A positive value gives two real roots. Zero gives one repeated root. A negative value gives two complex roots.
Can this calculator solve complex roots?
Yes. When the discriminant is negative, the calculator shows the real part and imaginary part of both complex roots.
What happens if a is zero?
If a is zero, the equation is not quadratic. The tool treats it as a linear equation when possible and shows a warning.
Why are exact roots helpful?
Exact roots avoid rounding errors. They are useful when answers need radicals or fractions instead of decimal approximations.
What is the vertex?
The vertex is the turning point of the parabola. Its x value is -b / 2a. The y value comes from substituting that x value.
Can I download my result?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a clean printable report.
How can I check the answer?
Substitute each root back into the original equation. The result should equal zero or be very close because of rounding.