Calculator
Enter a quadratic of the form a2 + bx + c, then convert it into vertex form.
Formula used
ax2 + bx + c
a(x - h)2 + k
h = -b / 2a
k = c - b2 / 4a
Divide the linear coefficient by the leading coefficient, take half of that value, square it, then add and subtract the same quantity.
How to use this calculator
- Enter coefficients a, b, and c from your quadratic expression.
- Pick the variable symbol and preferred decimal precision.
- Optionally enter a sample x value for quick function evaluation.
- Leave graph limits blank to let the page choose a useful viewing range.
- Press Complete the Square to see the transformed expression, steps, roots, and graph.
- Use the CSV or PDF buttons to save your working result.
Example data table
| a | b | c | Standard form | Vertex form | Vertex |
|---|---|---|---|---|---|
| 1 | 6 | 5 | x2 + 6x + 5 | (x + 3)2 - 4 | (-3, -4) |
| 2 | -8 | 3 | 2x2 - 8x + 3 | 2(x - 2)2 - 5 | (2, -5) |
| 1 | -4 | 1 | x2 - 4x + 1 | (x - 2)2 - 3 | (2, -3) |
| 3 | 12 | 7 | 3x2 + 12x + 7 | 3(x + 2)2 - 5 | (-2, -5) |
FAQs
1. What does completing the square mean?
It rewrites a quadratic into vertex form. That makes the shift, stretch, axis of symmetry, and turning point easier to identify directly.
2. Why must coefficient a be nonzero?
If a equals zero, the expression is no longer quadratic. Completing the square only applies to second-degree expressions with a squared term present.
3. What information does vertex form reveal?
Vertex form shows the vertex, axis of symmetry, opening direction, and vertical stretch immediately. It also helps graph the parabola quickly.
4. Does this calculator work with negative leading coefficients?
Yes. A negative a value means the parabola opens downward. The completion process is still valid, and the graph updates automatically.
5. What if the quadratic has no real roots?
The calculator still completes the square and reports complex conjugate roots. The graph remains helpful because it shows why the curve misses the x-axis.
6. Why is half of the linear coefficient important?
That half-value creates a perfect-square trinomial when squared. It is the key move that converts the normalized expression into a square plus or minus a constant.
7. Can I use decimals instead of integers?
Yes. Decimal coefficients work normally. The precision option helps you control rounding when the vertex, roots, or intermediate values are not whole numbers.
8. Why download the CSV or PDF result?
Exports are useful for homework review, tutoring notes, worksheet preparation, and checking multiple problems without recalculating everything from scratch later.