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Vertex Form Calculator (y = a(x − h)² + k)

Solve ax² + bx + c = 0 with the full quadratic formula worked out step by step: exact roots like (3 ± √5)/2 or −1 ± 2i, the discriminant, the vertex and the factored form.

Solves ax² + bx + c = 0 (a ≠ 0). Decimals and negative numbers are allowed.

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Reading coefficients
Computing the discriminant
Applying the quadratic formula
Done

Result

This page opens the quadratic equation solver with the vertex card put first: enter the coefficients of y = ax² + bx + c and the tool computes the vertex at h = −b/(2a) and k = −D/(4a) as exact reduced fractions, then writes the vertex form y = a(x − h)² + k with your numbers in it.

The vertex is where the parabola turns — its minimum when a > 0 and its maximum when a < 0 — so this page answers optimization and graphing questions directly: the axis of symmetry is x = h and the extreme value is k. For y = x² − 3x + 2 the vertex is (3/2, −1/4) and the vertex form is y = (x − 3/2)² − 1/4.

The step list shows how both coordinates come from the same two quantities you already know — b/2a and the discriminant — and the run also produces the roots and the factored form, which together give a complete sketch of the parabola: where it turns, where it crosses the x-axis, and how it factors.

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Accepted Formats & Rules

  • Handles all three cases: two real roots, one repeated root and complex conjugate roots.
  • Simplifies radicals (√48 → 4√3) and cancels common factors with 2a for the cleanest exact form.
  • Also reports the vertex (−b/2a, −D/4a) with the vertex form and factors the trinomial when the roots are rational.

Key Benefits

✓ Every equation is solved the way a teacher writes it: discriminant, substitution into the formula, radical simplification and the final roots as numbered steps.
✓ Exact symbolic answers — rational roots come out as reduced fractions, irrational ones as simplified radicals like (1 ± √5)/2, complex ones as a ± bi.
✓ Runs entirely in your browser: no sign-up, no uploads, instant results on any device.

Use Cases

  • • Solve any ax² + bx + c = 0 for homework with the full quadratic-formula working: discriminant, substitution, simplified radical and exact roots like (1 ± √5)/2.
  • • Check whether a trinomial factors: rational roots produce the integer factorization, such as (2x − 1)(3x − 1) for 6x² − 5x + 1.
  • • Find the vertex and axis of symmetry of a parabola exactly — h = −b/2a and k = −D/4a as reduced fractions with the vertex form written out.