Calculator Inputs
Formula Used
The standard form of a quadratic equation is:
ax² + bx + c = 0
The discriminant is:
D = b² - 4ac
The roots are calculated by:
x = (-b ± √D) / 2a
The vertex is:
x = -b / 2a and y = f(x)
The completed square form is based on:
a(x - h)² + k = 0
How To Use This Calculator
- Select full equation mode or coefficient mode.
- Enter an equation such as 2x^2 + 5x - 3 = 0.
- Or enter values for a, b, and c.
- Press the calculate button.
- Review the standard form and all advanced results.
- Use the CSV or PDF button to save your report.
Example Data Table
| Input Equation | Standard Form | Discriminant | Root Type |
|---|---|---|---|
| 2x^2 + 5x - 3 = 0 | 2x² + 5x - 3 = 0 | 49 | Two distinct real roots |
| x^2 - 4x + 4 = 0 | x² - 4x + 4 = 0 | 0 | One repeated real root |
| x^2 + 2x + 5 = 0 | x² + 2x + 5 = 0 | -16 | Two complex roots |
About This Standard Form Tool
A quadratic equation often appears in many shapes. It may be written as a product, a shifted expression, or a mixed equation with terms on both sides. This calculator rewrites the expression into standard form, ax² + bx + c = 0. It then solves the equation and explains the main values.
Why Standard Form Matters
Standard form gives every term a fixed place. The squared term comes first. The linear term follows. The constant stays last. Once the equation is arranged this way, the coefficients a, b, and c become easy to read. Those three values control the graph, roots, vertex, opening direction, and intercepts.
Advanced Results Included
The calculator checks the discriminant first. This value shows whether the roots are real, repeated, or complex. It also finds the axis of symmetry, vertex point, y-intercept, and x-intercepts. When possible, it shows an approximate factor form. It also gives a completed square form, which helps users understand vertex form.
Useful For Algebra Work
This tool is helpful when checking homework, building examples, or preparing lesson material. Students can enter coefficients directly. They can also type a full equation, such as 2x^2 + 5x - 3 = 0. The result area appears above the form after submission, so the answer is visible before editing values.
Export And Review
CSV export is useful for spreadsheets. PDF export is useful for saving a clean report. Both downloads include the main inputs and calculated results. The example table also shows common cases, including two real roots, one repeated root, and complex roots.
Because every calculation depends on accurate input, always check signs carefully. A missing minus sign can change the roots and vertex. For best results, write powers as x^2 and keep multiplication simple. The calculator is designed for educational use and transparent algebra review.
Interpreting The Graph
The sign of a tells the opening direction. A positive value opens upward. A negative value opens downward. The vertex gives the turning point. The axis passes through that point. The intercepts show where the graph meets each axis. These details connect algebra results with the shape of the parabola and support clearer problem checking. This makes each final answer easier to verify before submission.
FAQs
What is quadratic equation standard form?
Standard form is ax² + bx + c = 0. The values a, b, and c are coefficients. The value a cannot be zero.
Can I enter a full equation?
Yes. You can enter expressions like 2x^2 + 5x - 3 = 0. The calculator moves all terms to one side.
Can I enter only coefficients?
Yes. Choose coefficient mode. Then enter values for a, b, and c. The calculator will build the standard form.
What does the discriminant show?
The discriminant shows the root type. A positive value gives two real roots. Zero gives one repeated root. A negative value gives complex roots.
What is the vertex?
The vertex is the turning point of the parabola. It is found using x = -b / 2a and then evaluating y.
Does this calculator show complex roots?
Yes. When the discriminant is negative, the calculator shows complex roots using the imaginary unit i.
What does completed square form mean?
Completed square form rewrites the quadratic as a shifted square. It helps show the vertex and graph movement clearly.
Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.