BYTETOOLS

Completing the Square Calculator

Convert ax² + bx + c into vertex form a(x − h)² + k with every intermediate step shown. Get the vertex coordinates and check your algebra homework fast.

y = 2x² + 8x + 3

(-2, -5)
Vertex
x = -2
Axis of symmetry
-5
Minimum value

Vertex form

y = 2(x + 2)² − 5

Step by step

  1. Start with y = 2x² + 8x + 3
  2. Factor 2 out of the x² and x terms: y = 2(x² + 4x) + 3
  3. Take half of the x coefficient: (4) ÷ 2 = 2, then square it: (2)² = 4
  4. Add and subtract 4 inside: y = 2(x² + 4x + 4 − 4) + 3
  5. The first three terms form a perfect square: y = 2(x + 2)² − 2 × 4 + 3
  6. Combine the constants: -8 + 3 = -5
  7. Vertex form: y = 2(x + 2)² − 5

Formulas used: h = −b / 2a = -2, k = c − b² / 4a = -5. Expanding the vertex form multiplies back to the original quadratic.

What is the Completing the Square Calculator?

The ByteTools Completing the Square Calculator rewrites any quadratic ax² + bx + c in vertex form a(x − h)² + k.

  • Converts standard form to vertex form a(x − h)² + k
  • Every intermediate algebra step written out
  • Reports vertex (h, k), min/max and axis of symmetry
  • Handles a ≠ 1, negatives and decimals
  • Expansion check: vertex form multiplied back to standard form
  • 100% private — runs entirely in your browser

How to use the Completing the Square Calculator

  1. 1

    Enter the coefficients a, b and c of ax² + bx + c.

  2. 2

    Read the vertex form a(x − h)² + k at the top.

  3. 3

    Follow the numbered steps showing how the square was completed.

  4. 4

    Note the vertex (h, k) and the axis of symmetry.

  5. 5

    Copy the result and steps with one click.

About the Completing the Square Calculator

The ByteTools Completing the Square Calculator rewrites any quadratic ax² + bx + c in vertex form a(x − h)² + k. It computes h = −b / 2a and k = c − b² / 4a, and — more importantly for students — prints every intermediate step: factoring out a, halving the x coefficient, adding and subtracting the square, and regrouping.

Because vertex form exposes the parabola's turning point directly, the tool also reports the vertex (h, k), whether it is a minimum or maximum, and the axis of symmetry x = h. It is designed for algebra students learning the technique and for anyone who needs vertex form quickly for graphing.

Everything runs 100% locally in your browser. No values are uploaded or stored, results update as you type, and the full worked solution can be copied with one click.

Frequently asked questions

How do you complete the square?

Factor the leading coefficient out of the x² and x terms, take half of the remaining x coefficient, square it, and add and subtract that square inside the bracket. The first three terms then collapse to a perfect square. For x² + 6x + 5, half of 6 is 3, 3² = 9, giving (x + 3)² − 9 + 5 = (x + 3)² − 4.

What are h and k in vertex form?

In a(x − h)² + k, the point (h, k) is the vertex of the parabola. The formulas are h = −b / 2a and k = c − b² / 4a. The sign matters: (x + 3)² means h = −3, because x + 3 = x − (−3). The calculator gets these signs right and shows them explicitly.

Why complete the square at all?

Vertex form reveals the maximum or minimum point instantly, makes graphing straightforward, and is how the quadratic formula itself is derived. It is also used to rewrite circle and conic equations and to evaluate certain integrals in calculus later on.

Does completing the square work when a is not 1?

Yes — you first factor a out of the x² and x terms only, complete the square inside the bracket, then multiply back through. For 2x² + 8x + 3 that gives 2(x + 2)² − 5. The tool shows this factoring step whenever a ≠ 1.

How is the vertex related to the axis of symmetry?

The axis of symmetry is the vertical line through the vertex, x = h = −b / 2a. The parabola is a mirror image of itself across this line, which is why the two roots (when they exist) sit at equal distances either side of it.

Related tools