Factoring Quadratic Formula Calculator

Enter coefficients to factor any quadratic with guided algebra steps and explanations. See integer factorization, rational factors, or symbolic root forms with clarity throughout. Instantly view roots, vertex, axis, and completed square representations for deeper understanding. Download CSV and PDF to document your work.

Input Coefficients

Tip: Use keyboard Tab to jump fields.

Example Data Table

Try these classic cases. Click to load values.

abcFactorizationRootsLoad
156(x + 2)(x + 3)-2, -3
1-3-10(x - 5)(x + 2)5, -2
25-3(2x - 1)(x + 3)0.5, -3
30-273(x - 3)(x + 3)3, -3
441(2x + 1)²-0.5 (double)
521Irreducible over realsComplex
Results
Quadratic: 1x² + 5x + 6
Enter values and press Compute to see detailed results.
Formula Used

Quadratic formula: x = \u2212b \u00B1 \u221A(b\u00B2 \u2212 4ac) \u2215 2a.

  • Discriminant: Δ = b² − 4ac determines number and type of roots.
  • Vertex: h = −b/(2a), k = f(h).
  • Completed square: ax² + bx + c = a(x − h)² + k.
  • Factoring over integers uses the AC method when possible.
How to Use
  1. Enter coefficients a, b, and c for your quadratic.
  2. Click Compute to calculate discriminant, roots, and forms.
  3. Review exact and decimal roots, and any integer factorization.
  4. Read derivation steps for learning or verification.
  5. Use Download CSV or Download PDF to export results.
FAQs

The polynomial has no real roots and cannot be factored over the reals. It factors over complex numbers as (x − (p+qi))(x − (p−qi)).

Yes. The calculator handles decimals. Exact factorizations over integers appear only when coefficients allow integer factorization.

When the discriminant is not a perfect square, roots are irrational. Factoring uses exact radical expressions for precision.

The vertex is (h,k) with h = −b/(2a) and k = f(h). The axis is the vertical line x = h.

If integers m,n,p,q satisfy (mx+n)(px+q) = ax²+bx+c, then integer factorization exists. This happens when suitable divisors meet m·p=a, n·q=c, m·q+n·p=b.

Yes. Decimal roots are shown with six places for readability. Use exact forms when precision matters.

Multiplying all coefficients by a nonzero constant leaves roots unchanged. It scales any integer factorization accordingly.

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