Completing the Square (vertex form) — the idea

Completing the square rewrites a quadratic from standard form ax² + bx + c into vertex form a(x − h)² + k. Vertex form makes the vertex and axis of symmetry obvious: the vertex is (h, k) and the axis is x = h.

• x² + bx becomes (x + b/2)² − (b/2)².
• You “add and subtract” the same square so the expression stays equal.

• 1) Factor out a from the x-terms: a(x² + (b/a)x) + c.
• 2) Take half of the x coefficient inside parentheses: (b/2a).
• 3) Add and subtract its square: (b/2a)².
• 4) Rewrite as a perfect square: a(x + b/2a)² + (c − b²/4a).
• 5) Read vertex: h = −b/(2a), k = c − b²/(4a).

Completing the Square FAQs

• What does “completing the square” mean?

  It means rewriting a quadratic expression so it contains a perfect-square trinomial like (x − h)². You do this by adding and subtracting the same value so the expression stays equal.

• Does this change the graph?

  No — it’s the same parabola, just written in a different form. Vertex form simply reveals the vertex and axis of symmetry immediately.

• What if a = 0?

  Then it’s not a quadratic. This calculator will tell you it’s a linear (or constant) expression instead.

• Why is h = −b/(2a)?

  When you complete the square, the x-term turns into (x − h)². Expanding gives x² − 2hx + h², so matching coefficients leads to −2ah = b, which simplifies to h = −b/(2a).

• Can I use this to solve a quadratic equation?

  Yes. If you set ax² + bx + c = 0, convert to vertex form and then isolate the square: a(x − h)² = −k. Then take square roots to solve for x.

Two examples you can verify by expanding

Example 1: Convert x² + 8x + 5.

• Half of 8 is 4, square is 16.
• x² + 8x + 5 = (x + 4)² − 16 + 5 = (x + 4)² − 11.
• Vertex is (−4, −11).

Example 2: Convert 2x² − 12x + 1.

• Factor 2: 2(x² − 6x) + 1.
• Half of −6 is −3; square is 9.
• 2[(x − 3)² − 9] + 1 = 2(x − 3)² − 17.
• Vertex is (3, −17).

Completing the Square — complete guide (with intuition)

“Completing the square” is one of those algebra moves that feels like a trick the first time you see it, but it’s actually a structured way to rewrite a quadratic so the important geometry pops out. A quadratic in standard form ax² + bx + c hides its vertex. You can still find the vertex, but you either have to memorize formulas or do extra steps. Vertex form a(x − h)² + k is the opposite: it shows the vertex immediately, and from that you can sketch the parabola, find maximums/minimums, and solve equations by taking square roots.

The big idea is this: a perfect square looks like (x + d)². When you expand it you get x² + 2dx + d². Notice the middle term coefficient is always 2d. So if you want an expression like x² + bx to become a perfect square, you should pick d = b/2. That’s why “take half the x coefficient” is the key move. The only catch is that once you add (b/2)², you changed the value of the expression — so you subtract the same amount to keep things equal. Add and subtract the same number = no net change, but now you have a clean square you can factor.

1) The simplest case: a = 1

Start with x² + bx + c. Ignore c for a moment and focus on the x-terms: x² + bx. Half of b is b/2, and its square is (b/2)². Then:

• x² + bx becomes x² + bx + (b/2)² − (b/2)².
• Group the first three terms: (x + b/2)².
• Combine the constants: c − (b/2)².

Final result: x² + bx + c = (x + b/2)² + (c − b²/4). That’s already vertex form with a = 1, h = −b/2, and k = c − b²/4.

2) The general case: a ≠ 1

If a is not 1, you do one extra step first: factor a out of the x-terms. This makes the coefficient of x² inside the parentheses equal to 1 (which is what we need for the perfect-square pattern).

From ax² + bx + c: a(x² + (b/a)x) + c. Now the inside is like the simple case with b replaced by b/a. Half of b/a is b/(2a), and the square is b²/(4a²).

• Add and subtract that square inside: a[x² + (b/a)x + (b/2a)² − (b/2a)²] + c.
• Rewrite as a square: a(x + b/2a)².
• Distribute a onto the subtraction: a(x + b/2a)² − a·(b²/4a²) + c.
• Simplify the constant: −b²/(4a) + c.

Final result: ax² + bx + c = a(x + b/2a)² + (c − b²/(4a)). Most textbooks prefer a(x − h)² + k, so we set h = −b/(2a) and k = c − b²/(4a).

3) Reading the graph instantly

Once you have a(x − h)² + k:

• The vertex is (h, k).
• The axis of symmetry is x = h.
• If a > 0 the parabola opens up (vertex is a minimum). If a < 0 it opens down (vertex is a maximum).
• The value of k shifts the graph up/down; h shifts it left/right.

4) Solving quadratics using the completed square

Completing the square is also a full solution method. If you have an equation ax² + bx + c = 0, convert it to vertex form: a(x − h)² + k = 0. Then isolate the square: a(x − h)² = −k → (x − h)² = −k/a. Finally take square roots: x − h = ±√(−k/a), so x = h ± √(−k/a). This is exactly where the quadratic formula comes from, which is why mastering this method gives you “why”, not just “what”.

5) Common mistakes (and how to avoid them)

• Forgetting to factor out a: If a ≠ 1, complete the square after factoring a from x-terms.
• Not balancing the add/subtract: You must add and subtract the same amount (or add inside and subtract outside).
• Sign errors on h: In (x − h)², the sign flips. If you get (x + 4)², then h = −4.
• Skipping the check: Expand your vertex form quickly to confirm it matches the original.

6) Why teachers love this method

Completing the square connects algebra to geometry. If you think of a square of side length x, its area is x². Adding a strip of area bx can be visualized as attaching rectangles around the square and then filling in a missing corner to form a larger perfect square. That “missing corner” corresponds to the (b/2)² term you add. This geometric picture is also the gateway to conic sections (rewriting equations to see circles and ellipses) and later optimization in calculus (finding min/max).

7) Quick practice prompts

• Rewrite x² − 10x + 9 in vertex form.
• Rewrite 3x² + 6x − 2 in vertex form.
• For each, identify vertex and axis of symmetry without graphing.

Use the calculator above to check your answers, then do the expansion check yourself by hand — that’s the fastest way to build confidence and speed.

How this calculator formats results

To keep the output readable and “homework friendly,” the calculator simplifies fractions whenever possible and shows both the symbolic vertex form and the numeric vertex (h, k). When decimals are needed, it also shows a rounded value. If your input contains decimals, the exact fraction form may not always be representable; in that case the calculator focuses on a clean decimal step display.

• Use fraction inputs when you want exact radicals and exact vertex.
• Use decimal inputs when you want quick approximations for graphing.