Quadratic Formula Calculator
Enter values for the equation ax² + bx + c = 0. The calculator solves real or complex roots.
Example Data Table
| a | b | c | Discriminant | Root Type | Roots |
|---|---|---|---|---|---|
| 1 | -5 | 6 | 1 | Two real roots | 3, 2 |
| 1 | 2 | 1 | 0 | Repeated root | -1, -1 |
| 1 | 2 | 5 | -16 | Complex roots | -1 + 2i, -1 - 2i |
Formula Used
The standard quadratic equation is:
ax² + bx + c = 0
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
The discriminant is:
D = b² - 4ac
If D is positive, there are two real roots. If D is zero, there is one repeated real root. If D is negative, there are two complex roots.
The vertex is found with:
x = -b / 2a
y = ax² + bx + c
How to Use This Calculator
- Enter the value of coefficient a.
- Enter the value of coefficient b.
- Enter the value of coefficient c.
- Add an optional x value if you want f(x).
- Select the decimal precision.
- Click the calculate button.
- Review the result shown above the form.
- Use CSV or PDF buttons to download the report.
Advanced Quadratic Formula Guide
Why Quadratic Equations Matter
A quadratic equation appears when a variable is squared. It is common in algebra, physics, business, construction, and computer models. The equation can describe motion, area, profit, height, curves, and many design problems. This calculator gives more than the final roots. It also explains the structure of the equation.
Understanding the Discriminant
The discriminant is the expression inside the square root. It controls the type of answer. A positive discriminant gives two real roots. A zero discriminant gives one repeated root. A negative discriminant gives complex roots. This single value helps students understand the behavior of the whole graph.
Reading the Vertex
The vertex is the turning point of the parabola. If coefficient a is positive, the curve opens upward. The vertex is then the minimum point. If coefficient a is negative, the curve opens downward. The vertex is then the maximum point. This detail helps with optimization and graph analysis.
Using Root Relationships
The calculator also shows the sum and product of roots. These values come from coefficient relationships. The sum equals negative b divided by a. The product equals c divided by a. These checks are useful when verifying manual solutions or comparing factorized forms.
Practical Study Benefits
A full solution is better than a single answer. Learners can see roots, vertex, axis, intercept, and evaluated function value together. Teachers can create examples quickly. Students can download results for notes. The table also shows how the discriminant changes the answer type. This makes the calculator useful for practice, homework, checking, and lesson planning.
FAQs
1. What is a quadratic equation?
A quadratic equation is a second-degree equation. Its standard form is ax² + bx + c = 0. The value of a cannot be zero.
2. What does the quadratic formula find?
It finds the roots or solutions of a quadratic equation. These roots are the x-values where the equation equals zero.
3. What is the discriminant?
The discriminant is b² - 4ac. It tells whether the equation has two real roots, one repeated root, or complex roots.
4. Can this calculator solve complex roots?
Yes. If the discriminant is negative, the calculator displays roots using the imaginary unit i.
5. Why can coefficient a not be zero?
If a is zero, the equation is no longer quadratic. It becomes a linear equation and needs a different solving method.
6. What is the vertex of a parabola?
The vertex is the turning point of the graph. It can be a minimum point or a maximum point.
7. What does the axis of symmetry mean?
The axis of symmetry is a vertical line through the vertex. It divides the parabola into two matching sides.
8. Can I download my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable report.