Inputs
Equation format: a x² + b x + c = 0 with a ≠ 0.
Results
| Quantity | Value |
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Example Data
| # | a | b | c | Notes | Action |
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Formula Used
Starting with \\( a x^2 + b x + c = 0 \\) and \\( a \\neq 0 \\).
- Normalize: divide by \\( a \\) to obtain \\( x^2 + \\tfrac{b}{a}x + \\tfrac{c}{a} = 0 \\).
- Move constant: \\( x^2 + \\tfrac{b}{a}x = -\\tfrac{c}{a} \\).
- Complete the square by adding \\( \\left(\\tfrac{b}{2a}\\right)^2 \\) to both sides.
- Now \\( \\left(x + \\tfrac{b}{2a}\\right)^2 = \\tfrac{b^2 - 4ac}{4a^2} = \\tfrac{\\Delta}{4a^2} \\), where \\( \\Delta = b^2 - 4ac \\).
- Taking square roots: \\( x + \\tfrac{b}{2a} = \\pm \\tfrac{\\sqrt{\\Delta}}{2a} \\).
- Hence \\( x = \\tfrac{-b \\pm \\sqrt{\\Delta}}{2a} \\). The completing-square path produces the standard quadratic formula.
How to Use
- Enter coefficients
a,b,cfor the equationa x² + b x + c = 0. - Pick the desired number of decimal places for rounded values.
- Enable “Show step-by-step” to see the full completing-the-square derivation.
- Optionally show vertex form and geometric features for extra interpretation.
- Toggle Exact mode for fractions/radicals and Symbolic factorization if integers.
- Click Compute. Results, steps, vertex, and factorization hints will appear.
- Use Download CSV or Download PDF to export your findings.
FAQs
If
a=0, the equation becomes linear, not quadratic. This tool warns you and avoids division by zero.
The discriminant \\(\\Delta=b^2-4ac\\) determines root nature: positive gives two real roots, zero gives a repeated real root, negative gives complex-conjugate roots.
It reveals structure: the vertex form, axis of symmetry, and geometric meaning. It also derives the quadratic formula cleanly.
Yes. When \\(\\Delta<0\\), roots are reported in \\(p \\pm qi\\) notation with the chosen rounding applied to real and imaginary parts.
Vertex form is \\(y=a(x-h)^2+k\\) where \\(h=-\\tfrac{b}{2a}\\) and \\(k=f(h)=c-\\tfrac{b^2}{4a}\\). It shows the turning point and axis of symmetry clearly.
Rounding influences equality checks and products. Increase decimal places to reduce rounding error in displayed roots or factorization hints.
When roots are real, we present a numeric hint of the form \\(a(x-r_1)(x-r_2)\\). Exact symbolic factorization is shown when coefficients are integers.