Calculator Input
Formula Used
A quadratic expression starts as ax² + bx + c. The completed square form is a(x + b / 2a)² + c - b² / 4a. The vertex form is also written as a(x - h)² + k, where h = -b / 2a and k = c - b² / 4a.
The discriminant is D = b² - 4ac. If D is positive, the equation has two real roots. If D is zero, it has one repeated root. If D is negative, the roots are complex.
How to Use This Calculator
Enter the values of a, b, and c from your quadratic. Choose the variable and decimal precision. Pick a work mode if you want expression facts, equation roots, or graph facts. Press the submit button. The result will appear above the form. You can then download the calculation as a CSV file or a PDF report.
Example Data Table
| a | b | c | Standard Form | Completed Square Form | Vertex |
|---|---|---|---|---|---|
| 1 | 6 | 5 | x² + 6x + 5 | (x + 3)² - 4 | (-3, -4) |
| 2 | -8 | 3 | 2x² - 8x + 3 | 2(x - 2)² - 5 | (2, -5) |
| -1 | 4 | 7 | -x² + 4x + 7 | -(x - 2)² + 11 | (2, 11) |
Complete the Square Calculator Guide
What This Tool Does
This calculator rewrites a quadratic into completed square form. It helps you move from standard form to vertex form with less manual work. Standard form is ax² + bx + c. Vertex form is a(x - h)² + k. The numbers h and k give the vertex directly.
Why Completing the Square Matters
Completing the square is useful in algebra, graphing, and equation solving. It shows how a parabola shifts left, right, up, or down. It also explains the maximum or minimum value of a quadratic. That makes the method helpful for classroom work and practical modeling.
Understanding the Output
The result box gives the completed square form first. It also gives the vertex, axis of symmetry, discriminant, roots, focus, and directrix. These details help you check both algebraic and graph features. The step list shows the main transformation.
Checking the Vertex
The vertex is the turning point of the parabola. When a is positive, the vertex gives the minimum value. When a is negative, the vertex gives the maximum value. The axis of symmetry passes through the vertex. Its equation is x = h.
Using Roots and Discriminant
The discriminant tells how many real roots exist. A positive value gives two real roots. A zero value gives one repeated root. A negative value means the graph does not cross the horizontal axis. The calculator reports this clearly.
Download Options
The CSV download is useful for spreadsheets. The PDF download is useful for printing or saving a neat report. Both options include the main results. You can keep the file with homework notes, tutoring records, or lesson examples.
FAQs
1. What does completing the square mean?
It means rewriting a quadratic so one part becomes a perfect square. This reveals the vertex and makes graph details easier to read.
2. What form does this calculator create?
It creates completed square form, often called vertex form. The structure is a(x - h)² + k, where the vertex is (h, k).
3. Can coefficient a be zero?
No. If a is zero, the expression is linear, not quadratic. The calculator requires a nonzero value for coefficient a.
4. Does the calculator show roots?
Yes. It uses the discriminant to decide whether roots are two real values, one repeated value, or complex values.
5. Why is the vertex important?
The vertex is the turning point of the parabola. It also gives the minimum or maximum value of the quadratic.
6. What is the axis of symmetry?
The axis of symmetry is the vertical line passing through the vertex. For vertex (h, k), the axis is x = h.
7. Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.
8. Is this useful for graphing?
Yes. Vertex form, axis, opening direction, focus, and directrix all help describe the parabola graph clearly.