Rewrite quadratics into completed square form accurately. See vertex, shifts, roots, and graph updates instantly. Built for practice, checking homework, and classroom demonstrations daily.
Enter coefficients for ax² + bx + c. The page returns vertex form, steps, roots, and a graph.
| a | b | c | Completed square form | Vertex |
|---|---|---|---|---|
| 1 | 6 | 5 | (x + 3)2 - 4 | (-3, -4) |
| 2 | 8 | 1 | 2(x + 2)2 - 7 | (-2, -7) |
| -3 | 12 | -1 | -3(x - 2)2 + 11 | (2, 11) |
| 0.5 | -4 | 3 | 0.5(x - 4)2 - 5 | (4, -5) |
Start: ax2 + bx + c
Factor a: a[x2 + (b/a)x] + c
Complete the square: a[(x + b/(2a))2 - (b/(2a))2] + c
Vertex form: a(x - h)2 + k
Where: h = -b/(2a), k = c - b2/(4a)
The method first factors out the leading coefficient from the x-terms. Then it adds and subtracts the same square value. This preserves equality while converting the quadratic into vertex form.
It rewrites a quadratic from standard form into vertex form. That makes the vertex, axis of symmetry, maximum or minimum value, and graph shifts much easier to read.
When a is not 1, you must factor it from the x² and x terms first. That keeps the algebra balanced before adding the square value.
In a(x - h)² + k, the vertex is (h, k). This point is the minimum when a is positive and the maximum when a is negative.
Yes. Decimal coefficients are accepted. The calculator formats the simplified result, graph, vertex, and roots using trimmed decimal output for readability.
Yes. If the discriminant is negative, the page displays complex roots using real and imaginary parts instead of limiting results to real numbers.
Standard form is common in textbooks and input problems. Vertex form is better for interpreting shifts, turning points, and graph shape quickly.
The graph shows the parabola, vertex, and symmetry line. It helps confirm whether the algebraic rewrite matches the visual behavior of the quadratic.
CSV works well for spreadsheets and quick records. PDF is useful for printing, sharing homework checks, or saving clean reference notes.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.