Factored Form to Vertex Form Converter
Transform quadratic equations from factored form to vertex form instantly. Enter coefficients to reveal vertex coordinates and parabola graph.
Enter Factored Form: a(x – r)(x – s)
Conversion Formulas and Steps
Factored form a(x – r)(x – s) expands to standard ax² + bx + c, where b = -a(r + s), c = a r s. Complete the square for vertex form.
- Expand: y = a x² – a(r + s) x + a r s
- h = (r + s)/2
- k = a (h – r)² + a (h – s)² / 2, or use k = c – b²/(4a)
- Vertex form: y = a(x – h)² + k
Popular Factored to Vertex Conversions
| Factored Form | Vertex (h, k) | Vertex Form |
|---|---|---|
| (x-1)(x-3) | (2, -1) | (x-2)² – 1 |
| 2(x+1)(x-2) | (0.5, -1.5) | 2(x-0.5)² – 1.5 |
| (x-4)² | (4, 0) | (x-4)² |
| -0.5(x+2)(x+2) | (-2, -2) | -0.5(x+2)² – 2 |
| x(x-5) | (2.5, -6.25) | (x-2.5)² – 6.25 |
Graph Comparison
Blue line shows parabola from factored form; red dot marks vertex from vertex form. Axis of symmetry at x = h.
Common Questions
What if roots are equal?
Vertex form simplifies to a(x – r)² + k, perfect square parabola.
No real roots?
Vertex below x-axis (k < 0); graph shows minimum above or touches.
Negative a?
Parabola opens down; vertex is maximum point.
How to verify?
Plug h into equation: y(h) = k matches vertex.
References
Omni Calculator. Vertex Form Calculator. https://www.omnicalculator.com/math/vertex-form
Expii. Converting Between Different Forms of a Quadratic. https://www.expii.com/t/converting-between-different-forms-of-a-quadratic-4852
Khan Academy. Quadratic Functions & Equations. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations