Convert a quadratic from standard form ax² + bx + c to vertex form by completing the square, with the vertex, axis of symmetry, and step-by-step algebra.
For x² + 6x + 5, half of 6 is 3, and 3² is 9. Rewrite as (x + 3)² − 4, so the vertex is (−3, −4).
Enter a quadratic to rewrite it.
How to use the completing the square calculator
Enter the coefficients a, b, and c from a quadratic in standard form. The calculator rewrites the expression in vertex form, identifies the vertex, and shows the algebra path so you can compare the result with your textbook or class notes.
Use coefficients from ax² + bx + c. For x² + 6x + 5, enter a = 1, b = 6, c = 5.
The answer is written as a(x − h)² + k, where (h, k) is the vertex and x = h is the axis of symmetry.
Completing the square formula
For ax² + bx + c, the vertex form is a(x − h)² + k, where h = −b ÷ 2a and k = c − b² ÷ 4a. This is equivalent to completing the square after factoring out a from the x² and x terms.
Worked example
Rewrite x² + 6x + 5. Half of 6 is 3, and 3² = 9. Add and subtract 9 inside the expression: x² + 6x + 9 − 9 + 5. The first three terms become (x + 3)², and the constants combine to −4. The vertex form is (x + 3)² − 4.
| Part | Value |
|---|---|
| h | −3 |
| k | −4 |
| Vertex | (−3, −4) |
| Axis of symmetry | x = −3 |
Why completing the square is useful
Completing the square makes the shape of a quadratic easier to read. Standard form is useful for coefficients and y-intercept, while vertex form immediately shows the turning point. It also connects to the quadratic formula, graphing parabolas, and solving equations that cannot be factored neatly.
Common mistakes
- Forgetting to factor out a when a is not 1.
- Using b ÷ 2 instead of b ÷ 2a for the inside shift.
- Mixing the inside sign with the vertex sign: (x + 3)² has h = −3.
- Dropping the outside constant k after forming the square.
- Trying to use a = 0, which is not a quadratic equation.
Completing the square calculator FAQs
What is vertex form?
Vertex form is a(x − h)² + k. It shows the vertex of the parabola as (h, k).
Can this solve a quadratic equation?
This page rewrites the quadratic in vertex form. To solve for roots, use the related quadratic formula calculator or set the vertex form equal to zero and continue solving.
What if a is negative?
The method still works. A negative a means the parabola opens downward, but the vertex formula and completed-square form remain valid.
Why is the sign inside parentheses opposite the vertex x-value?
Vertex form uses x − h. If the form is (x + 3)², that is the same as x − (−3), so h = −3.
Is completing the square the same as factoring?
No. Factoring rewrites a quadratic as a product when possible. Completing the square rewrites it as a squared binomial plus or minus a constant.
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Reviewer
This educational math content is reviewed under the SolverCalc reviewer rule by Prof. Dr. Khalil Mudassar. WordPress adds the author bio automatically, so this page does not include a duplicate author box.
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