Completing the Square
Rewrite ax² + bx + c as a(x − h)² + k — the vertex form — with every step shown.
1x² + 6x + 5
Vertex (h, k) = (−3, −4) — the axis of symmetry is x = −3.
Completing the square rewrites ax² + bx + c as a(x − h)² + k, where h = −b ÷ 2a and k = c − b² ÷ 4a. For x² + 6x + 5, h = −3 and k = −4, giving (x + 3)² − 4. This vertex form reveals the vertex directly: the turning point is (−3, −4).
What completing the square does
Completing the square takes a quadratic written in standard form, ax² + bx + c, and rewrites it as a perfect square plus a constant: a(x − h)² + k. The two forms describe the same parabola, but vertex form hands you the vertex (h, k) with no extra work — and that makes graphing, solving, and finding maxima or minima far easier.
the vertex sits at (h, k); the axis of symmetry is the vertical line x = h
Worked example
Rewrite x² + 6x + 5 in vertex form, so a = 1, b = 6, c = 5:
- 1 Identify a, b, and c. For x² + 6x + 5 the coefficients are a = 1, b = 6, c = 5. Keep a factored out front if a ≠ 1.
- 2 Find h = −b ÷ 2a. h = −6 ÷ (2×1) = −3. This is the x-coordinate of the vertex and the axis of symmetry.
- 3 Find k = c − b² ÷ 4a. k = 5 − 6² ÷ (4×1) = 5 − 36 ÷ 4 = 5 − 9 = −4. This is the y-coordinate of the vertex.
- 4 Assemble a(x − h)² + k. With h = −3 and k = −4: (x − (−3))² − 4 = (x + 3)² − 4. Check: (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5. ✓
Reading the vertex form a(x − h)² + k
Every piece of the vertex form has a direct geometric meaning.
| Symbol | How to get it | What it tells you |
|---|---|---|
| h | −b ÷ 2a | x-coordinate of the vertex; the axis of symmetry is x = h |
| k | c − b² ÷ 4a | y-coordinate of the vertex; the minimum (a > 0) or maximum (a < 0) value |
| (h, k) | combine the two above | the vertex — the turning point of the parabola |
| a | the leading coefficient | opens upward if a > 0, downward if a < 0; sets the width |
| √(−k ÷ a) | solve a(x − h)² + k = 0 | the distance from h to each root — the quadratic formula in disguise |
Why it matters
Completing the square is the method that derives the quadratic formula: apply it to the general equation ax² + bx + c = 0 and solving a(x − h)² + k = 0 for x gives x = (−b ± √(b² − 4ac)) ÷ 2a. So every quadratic you solve with the formula is really a completed square underneath.
It also exposes the geometry at a glance. The vertex (h, k) is the parabola’s turning point, the line x = h is its axis of symmetry, and k is the extreme value — a minimum when a > 0 and a maximum when a < 0. The method only works when a ≠ 0; with a = 0 the expression is linear, not a parabola, and has no vertex.