Enter Quadratic Coefficients
Example Data Table
| a | b | c | Discriminant | Roots | Rational? |
|---|---|---|---|---|---|
| 1 | -5 | 6 | 1 | 2, 3 | Yes |
| 2 | 4 | 2 | 0 | -1, -1 | Yes |
| 3 | -2 | -1 | 16 | 1, -1/3 | Yes |
| 1 | 2 | 5 | -16 | -1 ± 2i | No |
Formula Used
Standard form: ax² + bx + c = 0
Discriminant: D = b² - 4ac
Quadratic formula: x = (-b ± √D) / 2a
Rational root test inside this calculator: roots are rational when D is a perfect square rational number.
Axis of symmetry: x = -b / 2a
Vertex y-value: y = c - b² / 4a
Sum of roots: -b / a
Product of roots: c / a
How to Use This Calculator
- Enter values for a, b, and c.
- Keep a different from zero.
- Use fractions like 5/2 when exact input matters.
- Choose decimal precision for rounded output.
- Set the x table range and step size.
- Press the calculate button.
- Read the discriminant, root type, exact roots, and factor form.
- Use CSV or PDF export for records.
Understanding Rational Quadratic Solutions
A Practical Root Check
A quadratic equation can model area, motion, revenue, and many classroom problems. It uses the form ax² + bx + c = 0. The value of a must not be zero. Otherwise, the expression becomes linear. This calculator focuses on exact rational solutions. A rational solution can be written as a fraction. That makes the answer easy to compare and verify.
Why the Discriminant Matters
The discriminant is the key test. It is found with b² - 4ac. A positive discriminant gives two real roots. A zero discriminant gives one repeated root. A negative discriminant gives complex roots. For rational roots, the discriminant must also be a perfect square rational number. The calculator checks that condition before labeling roots rational.
Exact Values and Decimal Values
Exact values are useful for algebra. Decimal values are useful for quick estimates. This tool shows both when possible. It also shows the formula form. That helps you follow the substitution process. If the roots are rational, the calculator reduces them. It also gives the sum and product of roots. These values help confirm the result.
Graph and Table Insight
The vertex gives the turning point of the parabola. The axis of symmetry passes through that point. The function table shows nearby values. This helps you see how the curve behaves. You can adjust the table range. Use a smaller step for more detail. Use a larger step for a quick overview.
Best Uses
Use this calculator for homework, lesson planning, checking factors, and reviewing exact roots. It is also helpful when coefficients are fractions. The export buttons save your work. The CSV file is useful for spreadsheets. The PDF file is useful for notes and printed review.
FAQs
1. What is a rational solution?
A rational solution is a root that can be written as a fraction. Examples include 3, -2, 1/4, and -7/5.
2. When does a quadratic have rational roots?
A quadratic has rational roots when its discriminant is a perfect square rational number. The coefficients must also form a valid quadratic equation.
3. What is the discriminant?
The discriminant is b² - 4ac. It tells whether the roots are real, repeated, complex, rational, or irrational.
4. Can I enter fractions?
Yes. You can enter values like 1/2, -3/4, or 7/5. The calculator reduces exact rational results.
5. What happens if a is zero?
The calculator shows an error. A zero value for a creates a linear equation, not a quadratic equation.
6. Why are some roots not rational?
Roots are not rational when the discriminant is not a perfect square rational number. They may be irrational or complex.
7. What does the vertex show?
The vertex shows the turning point of the parabola. It helps identify the minimum or maximum value of the quadratic function.
8. Why export results?
Exports help save calculations for homework, reports, classroom examples, and later checking. CSV works well for tables and spreadsheets.