Calculator inputs
Choose an input style, enter your values, and convert the equation into standard form ax² + bx + c = 0.
Formula used
Standard form target: ax² + bx + c = 0
From vertex form: a(x - h)² + k = r becomes ax² - 2ahx + (ah² + k - r) = 0
From factored form: a(x - r₁)(x - r₂) = r becomes ax² - a(r₁ + r₂)x + (ar₁r₂ - r) = 0
From balanced expanded form: a₁x² + b₁x + c₁ = a₂x² + b₂x + c₂ becomes (a₁ - a₂)x² + (b₁ - b₂)x + (c₁ - c₂) = 0
Discriminant: D = b² - 4ac
Roots: x = (-b ± √D) / 2a, when a is not zero.
How to use this calculator
- Select the equation format you already have.
- Enter the coefficients, shifts, factors, or both-side values.
- Click Convert to Standard Form.
- Read the generated ax² + bx + c = 0 result.
- Check the steps, discriminant, roots, and graph.
- Download the summary as CSV or PDF if needed.
Example data table
| Mode | Input | Converted standard form | A, B, C |
|---|---|---|---|
| Vertex | 2(x - 3)² - 5 = 0 | 2x² - 12x + 13 = 0 | 2, -12, 13 |
| Factored | 3(x - 4)(x + 1) = 0 | 3x² - 9x - 12 = 0 | 3, -9, -12 |
| Balanced expanded | x² + 7x + 10 = 2x - 4 | x² + 5x + 14 = 0 | 1, 5, 14 |
| Vertex with rhs | (x + 2)² + 7 = 3 | x² + 4x + 8 = 0 | 1, 4, 8 |
FAQs
1. What is standard form for a quadratic equation?
Standard form is ax² + bx + c = 0, where a cannot be zero. It collects every term on one side, making coefficients easy to read and compare.
2. Why convert a quadratic into standard form?
Standard form helps you identify coefficients quickly, compute the discriminant, apply the quadratic formula, graph the parabola, and compare equations using one consistent layout.
3. Can this calculator start from vertex form?
Yes. Enter a, h, k, and the right-side value. The calculator expands the square, distributes the leading coefficient, and rearranges the equation into standard form.
4. Can it also convert factored form?
Yes. It multiplies the two factors, applies the outside coefficient, then moves any right-side value so the final equation equals zero.
5. What does the discriminant tell me?
The discriminant, b² - 4ac, shows the root type. Positive means two real roots, zero means one repeated root, and negative means complex roots.
6. What happens if the x² term disappears?
If the final a coefficient becomes zero, the equation is no longer quadratic. The calculator still reports the simplified result and labels it correctly.
7. What do the CSV and PDF files contain?
They include the summary metrics shown on the result table, such as the original equation, standard form, coefficients, discriminant, roots, and graph-ready details.
8. Does the graph always show a parabola?
Only when the converted equation is truly quadratic. If the x² coefficient becomes zero, the plot becomes a line or constant level instead.